Areas by integration

209 questions · 25 question types identified

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Area between curve and line

A question is this type if and only if it requires finding the exact area between a curve and a non-horizontal straight line (such as a tangent or normal), requiring integration of the difference.

44 Standard +0.1
21.1% of questions
1.08e Area between curve and x-axis: using definite integrals
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\includegraphics{figure_1} The diagram shows the curve \(y = 6x - x^2\) and the line \(y = 5\). Find the area of the shaded region. You must show detailed reasoning. [4]
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Easiest question Moderate -0.8 »
4 The curve with equation \(y = x ^ { 4 } - 8 x + 9\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-5_410_609_383_721} The point \(( 2,9 )\) lies on the curve.
    1. Find \(\int _ { 0 } ^ { 2 } \left( x ^ { 4 } - 8 x + 9 \right) \mathrm { d } x\).
    2. Hence find the area of the shaded region bounded by the curve and the line \(y = 9\).
  2. The point \(A ( 1,2 )\) lies on the curve with equation \(y = x ^ { 4 } - 8 x + 9\).
    1. Find the gradient of the curve at the point \(A\).
    2. Hence find an equation of the tangent to the curve at the point \(A\).
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Hardest question Challenging +1.2 »
  1. Show that the point \(P ( 1,0 )\) lies on \(C\) .
  2. Find the coordinates of the point \(Q\) .
  3. Find the area of the shaded region between \(C\) and the line \(P Q\) .
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Area under polynomial curve

Find the exact area of a region bounded by a polynomial curve (integer powers only), the x-axis, and vertical lines, using direct integration.

28 Moderate -0.4
13.4% of questions
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Find the area of the finite region enclosed by the curve \(y = 5x - x^2\) and the \(x\)-axis. [6]
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Easiest question Easy -1.8 »
The graph of \(y = x^2 - 9\) is shown below. \includegraphics{figure_1} Find the area of the shaded region. Circle your answer. [1 mark] \(-18\) \quad\quad \(-6\) \quad\quad \(6\) \quad\quad \(18\)
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Hardest question Challenging +1.2 »
\includegraphics{figure_8} The diagram shows the curve with equation \(y = 5x^4 + ax^3 + bx\), where \(a\) and \(b\) are integers. The curve has a minimum at the point \(P\) where \(x = 2\). The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = 2\). Given that the area of the shaded region is \(48\) units\(^2\), determine the \(y\)-coordinate of \(P\). [7]
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Combined region areas

A question is this type if and only if it asks to find areas of multiple separate regions and combine them, or find total shaded area consisting of distinct parts.

24 Standard +0.1
11.5% of questions
1.08e Area between curve and x-axis: using definite integrals
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The graph of \(y = x^3\) is shown. \includegraphics{figure_1} Find the total shaded area. Circle your answer. [1 mark] \(-68\) 60 68 128
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Easiest question Easy -1.8 »
The graph of \(y = x^3\) is shown. \includegraphics{figure_1} Find the total shaded area. Circle your answer. [1 mark] \(-68\) 60 68 128
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Hardest question Challenging +1.2 »
\includegraphics{figure_5} Figure 5 shows a sketch of part of the curve \(y = 2x + \frac{8}{x^2} - 5\), \(x > 0\). The point \(A(4, \frac{7}{2})\) lies on C. The line \(l\) is the tangent to C at the point A. The region \(R\), shown shaded in figure 5 is bounded by the line \(l\), the curve C, the line with equation \(x = 1\) and the \(x\)-axis. Find the exact area of \(R\). (Solutions based entirely on graphical or numerical methods are not acceptable.)
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Deduce related integral from numerical approximation

A question that asks to use a trapezium rule or Simpson's rule result to deduce the value of a related integral through algebraic manipulation (e.g. using log laws or scalar multiples to transform the estimated integral).

23 Standard +0.0
11.0% of questions
1.09f Trapezium rule: numerical integration
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2
  1. Use Simpson's rule with four strips to find an approximation to $$\int _ { 4 } ^ { 12 } \ln x \mathrm {~d} x$$ giving your answer correct to 2 decimal places.
  2. Deduce an approximation to \(\int _ { 4 } ^ { 12 } \ln \left( x ^ { 10 } \right) \mathrm { d } x\).
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Easiest question Moderate -0.8 »
  1. (a) Given that \(a\) is a constant, \(a > 1\), sketch the graph of
$$y = a ^ { x } , \quad x \in \mathbb { R }$$ On your diagram show the coordinates of the point where the graph crosses the \(y\)-axis.
(2) The table below shows corresponding values of \(x\) and \(y\) for \(y = 2 ^ { x }\)
\(x\)- 4- 2024
\(y\)0.06250.251416
(b) Use the trapezium rule, with all of the values of \(y\) from the table, to find an approximate value, to 2 decimal places, for $$\int _ { - 4 } ^ { 4 } 2 ^ { x } \mathrm {~d} x$$ (c) Use the answer to part (b) to find an approximate value for
  1. \(\int _ { - 4 } ^ { 4 } 2 ^ { x + 2 } \mathrm {~d} x\)
  2. \(\int _ { - 4 } ^ { 4 } \left( 3 + 2 ^ { x } \right) \mathrm { d } x\)
    \includegraphics[max width=\textwidth, alt={}, center]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-23_86_47_2617_1886}
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Hardest question Standard +0.8 »
6. (i) Write down the exact value of \(\cos \frac { \pi } { 6 }\). The finite region \(R\) is bounded by the curve \(y = \cos ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(ii) Use the trapezium rule with two intervals of equal width to estimate the area of \(R\), giving your answer to 3 significant figures. The finite region \(S\) is bounded by the curve \(y = \sin ^ { 2 } x\), where \(x\) is measured in radians, the positive coordinate axes and the line \(x = \frac { \pi } { 3 }\).
(iii) Using your answer to part (b), find an estimate for the area of \(S\).
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Area of sector/segment problems

A question is this type if and only if it involves finding areas of circular sectors, segments, or regions bounded by circular arcs and straight lines, using geometric formulas.

21 Standard +0.3
10.0% of questions
1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta
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3 In the diagram, \(A C\) is an arc of a circle, centre \(O\) and radius 6 cm . The line \(B C\) is perpendicular to \(O C\) and \(O A B\) is a straight line. Angle \(A O C = \frac { 1 } { 3 } \pi\) radians. Find the area of the shaded region, giving your answer in terms of \(\pi\) and \(\sqrt { } 3\).
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Easiest question Moderate -0.5 »
5 \includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-06_323_775_260_685} The diagram shows a triangle \(O A B\) in which angle \(O A B = 90 ^ { \circ }\) and \(O A = 5 \mathrm {~cm}\). The arc \(A C\) is part of a circle with centre \(O\). The arc has length 6 cm and it meets \(O B\) at \(C\). Find the area of the shaded region.
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Hardest question Challenging +1.2 »
11 \includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-4_611_668_1699_737} The diagram shows a sector of a circle with centre \(O\) and radius 20 cm . A circle with centre \(C\) and radius \(x \mathrm {~cm}\) lies within the sector and touches it at \(P , Q\) and \(R\). Angle \(P O R = 1.2\) radians.
  1. Show that \(x = 7.218\), correct to 3 decimal places.
  2. Find the total area of the three parts of the sector lying outside the circle with centre \(C\).
  3. Find the perimeter of the region \(O P S R\) bounded by the \(\operatorname { arc } P S R\) and the lines \(O P\) and \(O R\).
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Area involving fractional powers

A question is this type if and only if the curve equation involves fractional or negative powers of x (like x^(1/2), x^(-2)) requiring power rule integration.

10 Moderate -0.4
4.8% of questions
1.08e Area between curve and x-axis: using definite integrals
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2 Find the area of the region enclosed by the curve \(y = 2 \sqrt { } x\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).
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Easiest question Easy -1.8 »
2 Find the area of the region enclosed by the curve \(y = 2 \sqrt { } x\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).
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Hardest question Standard +0.3 »
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{089f5506-94ac-489f-b219-e67fa6ca834f-4_536_883_248_486} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the curve with equation \(y = \sqrt { x } + \frac { 8 } { x ^ { 2 } } , x > 0\).
  1. Find the coordinates of the minimum point of the curve.
  2. Show that the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 9\) is \(24 \frac { 4 } { 9 }\).
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Area under curve with fractional/negative powers or roots

Find the exact area of a region bounded by a curve involving fractional powers, negative powers, or square roots, the x-axis, and vertical lines, using direct integration.

10 Standard +0.1
4.8% of questions
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\includegraphics{figure_1} The diagram shows the curve \(y = \sqrt{x - 3}\). The shaded region is bounded by the curve and the two axes. Find the exact area of the shaded region. [4]
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Easiest question Moderate -0.3 »
\includegraphics{figure_5} The diagram shows the curve with equation \(y = 10x^{\frac{1}{2}} - \frac{5}{2}x^{\frac{3}{2}}\) for \(x > 0\). The curve meets the \(x\)-axis at the points \((0, 0)\) and \((4, 0)\). Find the area of the shaded region. [4]
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Hardest question Standard +0.8 »
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a377da06-a968-438c-bec2-ae55283dae47-28_533_1095_258_365} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Diagram not drawn to scale
  1. Find $$\int \frac { 1 } { ( 2 x - 1 ) ^ { 2 } } d x$$ Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 12 } { ( 2 x - 1 ) } \quad 1 \leqslant x \leqslant 5$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the line with equation \(x = 1\), the curve with equation \(y = \mathrm { f } ( x )\) and the line with equation \(y = \frac { 4 } { 3 }\). The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  2. Find the exact value of the volume of the solid generated, giving your answer in its simplest form.
    \section*{Leave
    k}
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Trapezium rule estimation

A question is this type if and only if it asks to use the trapezium rule to estimate an area or integral value, typically requiring completion of a table and/or calculation with given ordinates.

5 Moderate -0.3
2.4% of questions
1.02l Modulus function: notation, relations, equations and inequalities1.09f Trapezium rule: numerical integration
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1 Use the trapezium rule with 3 intervals to estimate the value of $$\int _ { 0 } ^ { 3 } \left| 2 ^ { x } - 4 \right| \mathrm { d } x$$
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Simpson's rule estimation

A question is this type if and only if it asks to use Simpson's rule (not trapezium rule) to estimate an integral value.

5 Standard +0.1
2.4% of questions
1.09f Trapezium rule: numerical integration
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The integral \(I\) is defined by $$I = \int_0^{13} (2x + 1)^{\frac{3}{2}} \, dx.$$
  1. Use integration to find the exact value of \(I\). [4]
  2. Use Simpson's rule with two strips to find an approximate value for \(I\). Give your answer correct to 3 significant figures. [3]
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Exact area with surds

A question is this type if and only if it requires finding an exact area that must be expressed in surd form (containing square roots) rather than decimals.

5 Moderate -0.1
2.4% of questions
1.08e Area between curve and x-axis: using definite integrals
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3 Show that the area of the region bounded by the curve \(y = 3 x ^ { - \frac { 3 } { 2 } }\), the lines \(x = 1 , x = 3\) and the \(x\)-axis is \(6 - 2 \sqrt { 3 }\).
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Region bounded by two curves

A question is this type if and only if it asks to find the area between two curves (not curve and line), requiring integration of the difference between two functions.

5 Moderate -0.1
2.4% of questions
1.08e Area between curve and x-axis: using definite integrals
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\includegraphics{figure_9} The diagram above shows a sketch of the curves \(y = x^2 + 4\) and \(y = 12 - x^2\). Find the area of the region bounded by the two curves. [6]
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Complete table then estimate

A question is this type if and only if it first requires completing missing values in a table of coordinates, then using those values with trapezium rule.

4 Moderate -0.1
1.9% of questions
1.09f Trapezium rule: numerical integration
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1. $$y = \sqrt { \left( 2 ^ { x } + x \right) }$$ a. Complete the table below, giving the values of \(y\) to 3 decimal places.
\(x\)00.20.40.60.81
\(y\)11.1611.3111.732
(1)
b. Use the trapezium rule with all the values of \(y\) from your table to find an approximation for the value of $$\int _ { 0 } ^ { 1 } \sqrt { \left( 2 ^ { x } + x \right) } \mathrm { d } x$$ giving your answer to 3 significant figures. Using your answer to part (b) and making your method clear, estimate
c. \(\int _ { 0 } ^ { 0.5 } \sqrt { \left( 2 ^ { 2 x } + 2 x \right) } \mathrm { d } x\)
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Area with parametric/substitution

A question is this type if and only if finding the area requires using a substitution method or parametric equations, not just direct integration.

3 Standard +0.3
1.4% of questions
1.08h Integration by substitution
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5. \includegraphics[max width=\textwidth, alt={}, center]{5840974b-b08a-4818-9a59-97b2d3ce9890-1_469_809_1777_484} The diagram shows the curve with equation \(y = x \sqrt { 1 - x } , 0 \leq x \leq 1\).
Use the substitution \(u ^ { 2 } = 1 - x\) to show that the area of the region bounded by the curve and the \(x\)-axis is \(\frac { 4 } { 15 }\).
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Bounds using rectangles

A question is this type if and only if it asks to find upper and lower bounds for an area by considering rectangles under or over a curve, often with summation notation.

3 Standard +0.6
1.4% of questions
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\includegraphics{figure_3} The diagram shows the curve with equation \(y = 1 - x^2\) for \(0 \leq x \leq 1\), together with a set of \(n\) rectangles of width \(\frac{1}{n}\).
  1. By considering the sum of the areas of the rectangles, show that $$\int_0^1 (1 - x^2) \, dx < \frac{4n^2 + 3n - 1}{6n^2}.$$ [4]
  2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int_0^1 (1 - x^2) \, dx\). [4]
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Area between two curves

Find the exact area of a region bounded by two curves (neither being a non-horizontal straight line), requiring integration of the difference of two functions.

3 Moderate -0.3
1.4% of questions
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\includegraphics{figure_9} The diagram shows the curves with equations \(y = x^3 - 3x + 3\) and \(y = 2x^3 - 4x^2 + 3\).
  1. Find the \(x\)-coordinates of the points of intersection of the curves. [3]
  2. Find the area of the shaded region. [4]
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Trapezium rule with reasoning

A question is this type if and only if it asks to use the trapezium rule AND requires explanation of whether the result is an overestimate or underestimate, or how to improve accuracy.

2 Easy -1.1
1.0% of questions
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The trapezium rule is used to estimate the area of the shaded region in each of the graphs below. Identify the graph for which the trapezium rule produces an overestimate. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_2}
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Trapezium rule for applications

A question is this type if and only if it uses the trapezium rule in a real-world context (river cross-section, tunnel volume, distance from speed, etc.) rather than pure mathematical area.

2 Moderate -0.2
1.0% of questions
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\includegraphics{figure_12} A water trough is a prism 2.5 m long. Fig. 12 shows the cross-section of the trough, with the depths in metres at 0.1 m intervals across the trough. The trough is full of water.
  1. Use the trapezium rule with 5 strips to calculate an estimate of the area of cross-section of the trough. Hence estimate the volume of water in the trough. [5]
  2. A computer program models the curve of the base of the trough, with axes as shown and units in metres, using the equation \(y = 8x^3 - 3x^2 - 0.5x - 0.15\), for \(0 \leq x \leq 0.5\). Calculate \(\int_0^{0.5} (8x^3 - 3x^2 - 0.5x - 0.15) \, \text{d}x\) and state what this represents. Hence find the volume of water in the trough as given by this model. [7]
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Area with logarithmic functions

A question is this type if and only if the curve equation involves logarithmic functions (ln x or log x) and requires integration techniques specific to logarithms.

2 Standard +0.2
1.0% of questions
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In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve \(y = \frac{1}{x+2}\), the two axes and the line \(x = 2.5\). [3]
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Area with trigonometric functions

A question is this type if and only if the curve equation involves trigonometric functions (sin, cos, tan, sec) and requires integration of trigonometric expressions.

2 Standard +0.8
1.0% of questions
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14 In this question you must show detailed reasoning. Fig. 14 shows the graphs of \(y = \sin x \cos 2 x\) and \(y = \frac { 1 } { 2 } - \sin 2 x \cos x\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cea67565-8074-4703-8e1a-09b98e380baf-16_647_898_404_233} \captionsetup{labelformat=empty} \caption{Fig. 14}
\end{figure} Use integration to find the area between the two curves, giving your answer in an exact form.
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Area under transcendental or composite curve

Find the exact area of a region bounded by a curve involving exponential, logarithmic, or trigonometric functions, the x-axis, and vertical lines, using direct integration.

2 Standard +0.8
1.0% of questions
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  1. Given that \(y = \frac{4 \ln x - 3}{4 \ln x + 3}\), show that \(\frac{dy}{dx} = \frac{24}{x(4 \ln x + 3)^2}\). [3]
  2. Find the exact value of the gradient of the curve \(y = \frac{4 \ln x - 3}{4 \ln x + 3}\) at the point where it crosses the \(x\)-axis. [4]
  3. \includegraphics{figure_8iii} The diagram shows part of the curve with equation $$y = \frac{2}{x^2(4 \ln x + 3)}.$$ The region shaded in the diagram is bounded by the curve and the lines \(x = 1\), \(x = e\) and \(y = 0\). Find the exact volume of the solid produced when this shaded region is rotated completely about the \(x\)-axis. [4]
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Area involving absolute values

A question is this type if and only if the integrand contains an absolute value function and requires consideration of where the function changes sign.

1 Standard +0.8
0.5% of questions
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  1. Sketch on the same diagram the graphs of \(y = |x| - a\) and \(y = |3x + 5a|\), where \(a\) is a positive constant. Show on your diagram the coordinates of any points where each graph meets the coordinate axes. [5]
  2. Solve the equation $$|x| - a = |3x + 5a|.$$ [4]
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Area with exponential functions

A question is this type if and only if the curve equation involves exponential functions (e^x or a^x) and requires integration of exponential expressions.

1 Standard +0.3
0.5% of questions
Error analysis for approximation

A question is this type if and only if it asks to calculate the actual error or percentage error between a trapezium rule estimate and the exact value found by integration.

1 Moderate -0.3
0.5% of questions
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The finite region \(R\) is bounded by the curve \(y = 1 + 3\sqrt{x}\), the \(x\)-axis and the lines \(x = 2\) and \(x = 8\).
  1. Use the trapezium rule with three intervals of equal width to estimate to 3 significant figures the area of \(R\). [6]
  2. Use integration to find the exact area of \(R\) in the form \(a + b\sqrt{2}\). [5]
  3. Find the percentage error in the estimate made in part (a). [2]
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Area under piecewise-defined curve

Find the exact area of a region bounded by a piecewise-defined function and the x-axis, requiring separate integration over each sub-interval and combining results.

1 Moderate -0.3
0.5% of questions
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11 The cross-section of a brick wall built on horizontal ground is given, for \(0 \leq x \leq 6\), by the following function $$\begin{array} { l l } 0 \leq x \leq 2 & y = 1 \\ 2 \leq x \leq 4 & y = - \frac { 1 } { 2 } x ^ { 2 } + 3 x - 3 \\ 4 \leq x \leq 6 & y = 1 \end{array}$$
\includegraphics[max width=\textwidth, alt={}]{13bfa97b-ec49-4f41-b3dd-d9a31a2c30e8-4_523_1327_633_413}
Units are metres.
  1. Show that the highest point on the wall is 1.5 metres above the ground.
  2. Find the area of the cross-section of the wall.
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Trapezium/Simpson's rule estimation

A question that asks only to apply the trapezium rule or Simpson's rule to estimate a definite integral numerically, without requiring deduction of a related integral.

0
0.0% of questions
Unclassified

Questions not yet assigned to a type.

2
1.0% of questions
Show 2 unclassified »
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c46ca445-cf59-4664-931e-add9f2f81851-08_419_665_255_708} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.} Figure 1 shows a sketch of the curve with equation $$y = \sqrt { \frac { 3 x } { 3 x ^ { 2 } + 5 } } \quad x \geqslant 0$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines with equations \(x = \sqrt { 5 }\) and \(x = 5\) The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
Use integration to find the exact volume of the solid generated. Give your answer in the form \(a \ln b\), where \(a\) is an irrational number and \(b\) is a prime number.
8
  1. Given that \(\mathrm { y } = \frac { 4 \ln \mathrm { x } - 3 } { 4 \ln \mathrm { x } + 3 }\), show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = \frac { 24 } { \mathrm { x } ( 4 \ln \mathrm { x } + 3 ) ^ { 2 } }\).
  2. Find the exact value of the gradient of the curve \(y = \frac { 4 \ln x - 3 } { 4 \ln x + 3 }\) at the point where it crosses the \(x\)-axis.
  3. \includegraphics[max width=\textwidth, alt={}, center]{133c38fb-307f-4f20-86cb-1bd57cc4f870-3_524_830_941_699} The diagram shows part of the curve with equation $$\mathrm { y } = \frac { 2 } { \mathrm { x } ^ { \frac { 1 } { 2 } } ( 4 \ln \mathrm { x } + 3 ) }$$ The region shaded in the diagram is bounded by the curve and the lines \(x = 1 , x = e\) and \(y = 0\). Find the exact volume of the solid produced when this shaded region is rotated completely about the x -axis.