Direct integration for area

9 \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-12_764_967_292_555} The diagram shows the curve with equation \(y = \sqrt { 2 x ^ { 3 } + 10 }\).

1. Find the equation of the tangent to the curve at the point where \(x = 3\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-12_2716_35_141_2013}
2. The region shaded in the diagram is enclosed by the curve and the straight lines \(x = 1 , x = 3\) and \(y = 0\). Find the volume of the solid obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

9 A curve has equation \(y = \frac { 4 } { \sqrt { } x }\).

1. The normal to the curve at the point \(( 4,2 )\) meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Find the length of \(P Q\), correct to 3 significant figures.
2. Find the area of the region enclosed by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 4\).

2 \includegraphics[max width=\textwidth, alt={}, center]{c8925c7a-cb3b-43b8-9d09-8adc800c6887-2_627_828_641_657} The diagram shows the curve \(y = 2 x ^ { 2 }\) and the points \(X ( - 2,0 )\) and \(P ( p , 0 )\). The point \(Q\) lies on the curve and \(P Q\) is parallel to the \(y\)-axis.

1. Express the area, \(A\), of triangle \(X P Q\) in terms of \(p\). The point \(P\) moves along the \(x\)-axis at a constant rate of 0.02 units per second and \(Q\) moves along the curve so that \(P Q\) remains parallel to the \(y\)-axis.
2. Find the rate at which \(A\) is increasing when \(p = 2\).

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation $$y = 2 x ^ { 2 } + 7 \quad x \geqslant 0$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(y\)-axis and the line with equation \(y = 17\) Find the exact area of \(R\).

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = ( x + 1 ) ( x - 5 )$$ The curve crosses the \(x\)-axis at the points \(A\) and \(B\).

1. Write down the \(x\)-coordinates of \(A\) and \(B\). The finite region \(R\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis.
2. Use integration to find the area of \(R\).

7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = 3 x - x ^ { \frac { 3 } { 2 } } , \quad x \geqslant 0$$ The finite region \(S\), bounded by the \(x\)-axis and the curve, is shown shaded in Figure 3.

1. Find $$\int \left( 3 x - x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x$$
2. Hence find the area of \(S\).

9. \begin{figure}[h] \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.
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\begin{figure}[h] \end{figure} Figure 1 shows part of the curve \(C\) with equation \(y = ( 1 + x ) ( 4 - x )\).
The curve intersects the \(x\)-axis at \(x = - 1\) and \(x = 4\). The region \(R\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis. Use calculus to find the exact area of \(R\).

10 \includegraphics[max width=\textwidth, alt={}, center]{dd199f4d-8cf3-4b1e-92aa-d54e9e94da57-4_693_931_269_607} The diagram shows the graph of \(y = 1 - 3 x ^ { - \frac { 1 } { 2 } }\).

1. Verify that the curve intersects the \(x\)-axis at \(( 9,0 )\).
2. The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = a\) (where \(a > 9\) ). Given that the area of the shaded region is 4 square units, find the value of \(a\).

11 The cross-section of a brick wall built on horizontal ground is given, for \(0 \leq x \leq 6\), by the following function $$\begin{array} { l l } 0 \leq x \leq 2 & y = 1 \\ 2 \leq x \leq 4 & y = - \frac { 1 } { 2 } x ^ { 2 } + 3 x - 3 \\ 4 \leq x \leq 6 & y = 1 \end{array}$$ 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.

1

1. Find \(\int \left( x ^ { 3 } - 2 x \right) \mathrm { d } x\). The graph below shows part of the curve \(y = x ^ { 3 } - 2 x\) for \(0 \leq x \leq 2\). \includegraphics[max width=\textwidth, alt={}, center]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-2_528_1019_520_321}
2. Show that the area of the shaded region \(P\) is the same as the area of the shaded region \(Q\).

6. \includegraphics[max width=\textwidth, alt={}, center]{38c285f8-a542-42fb-aa4c-41adb9a8f63e-2_503_839_1208_557} The diagram shows the curve with equation \(y = 4 x ^ { \frac { 1 } { 3 } } - x , x \geq 0\).
The curve meets the \(x\)-axis at the origin and at the point \(A\) with coordinates \(( a , 0 )\).

1. Show that \(a = 8\).
2. Find the area of the finite region bounded by the curve and the positive \(x\)-axis.

9. \includegraphics[max width=\textwidth, alt={}, center]{e4afa57d-5be3-42a6-ab35-39b0fdcc1681-3_559_732_824_388} The diagram shows the curve with equation \(y = \mathrm { f } ( x )\) which crosses the \(x\)-axis at the origin and at the points \(A\) and \(B\). Given that $$f ^ { \prime } ( x ) = 4 - 6 x - 3 x ^ { 2 }$$

1. find an expression for \(y\) in terms of \(x\),
2. show that \(A\) has coordinates ( \(- 4,0\) ) and find the coordinates of \(B\),
3. find the total area of the two regions bounded by the curve and the \(x\)-axis.

4 The equation of a curve is \(y = 9 x ^ { 2 } - x ^ { 4 }\).

1. Show that the curve meets the \(x\)-axis at the origin and at \(x = \pm a\), stating the value of \(a\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\). Hence show that the origin is a minimum point on the curve. Find the \(x\)-coordinates of the maximum points.
3. Use calculus to find the area of the region bounded by the curve and the \(x\)-axis between \(x = 0\) and \(x = a\), using the value you found for \(a\) in part (i).

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.

6

1. Use integration to find the exact area of the region enclosed by the curve \(y = x ^ { 2 } + 4 x\), the \(x\)-axis and the lines \(x = 3\) and \(x = 5\).
2. Find \(\int ( 2 - 6 \sqrt { y } ) \mathrm { d } y\).
3. Evaluate \(\int _ { 1 } ^ { \infty } \frac { 8 } { x ^ { 3 } } \mathrm {~d} x\).

12 The equation of a curve is \(y = 9 x ^ { 2 } - x ^ { 4 }\).

1. Show that the curve meets the \(x\)-axis at the origin and at \(x = \pm a\), stating the value of \(a\).
2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\). Hence show that the origin is a minimum point on the curve. Find the \(x\)-coordinates of the maximum points.
3. Use calculus to find the area of the region bounded by the curve and the \(x\)-axis between \(x = 0\) and \(x = a\), using the value you found for \(a\) in part (i).

1 The graph of \(y = x ^ { 2 } - 9\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{be9ad375-d3cd-488c-9e3b-a4cd035d0d1d-02_335_593_767_744} Find the area of the shaded region.
Circle your answer.
[0pt] [1 mark] \(- 18 - 6618\)

13. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = 2 x ^ { 3 } - 17 x ^ { 2 } + 40 x$$ The curve has a minimum turning point at \(x = k\).
The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = k\). Show that the area of \(R\) is \(\frac { 256 } { 3 }\) (Solutions based entirely on graphical or numerical methods are not acceptable.)

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} The finite region \(R\), shown shaded in Figure 2, is bounded by the curve with equation \(y = 4 x ^ { 2 } + 3\), the \(y\)-axis and the line with equation \(y = 23\) Show that the exact area of \(R\) is \(k \sqrt { 5 }\) where \(k\) is a rational constant to be found.

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \sqrt { x } , x \geqslant 0\) The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line with equation \(x = 1\), the \(x\)-axis and the line with equation \(x = a\), where \(a\) is a constant. Given that the area of \(R\) is 10

1. find, in simplest form, the value of
   1. \(\int _ { 1 } ^ { a } \sqrt { 8 x } \mathrm {~d} x\)
   2. \(\int _ { 0 } ^ { a } \sqrt { x } \mathrm {~d} x\)
2. show that \(a = 2 ^ { k }\), where \(k\) is a rational constant to be found.

15. \begin{figure}[h] \end{figure} Figure 7 shows an open tank for storing water, \(A B C D E F\). The sides \(A C D F\) and \(A B E F\) are rectangles. The faces \(A B C\) and \(F E D\) are sectors of a circle with radius \(A B\) and \(F E\) respectively.

• \(A B = F E = r \mathrm {~cm}\)
• \(A F = B E = C D = l \mathrm {~cm}\)
• angle \(B A C =\) angle \(E F D = 0.9\) radians

Given that the volume of the tank is \(360 \mathrm {~cm} ^ { 3 }\) a. show that the surface area of the tank, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 0.9 r ^ { 2 } + \frac { 1600 } { r }$$ (4) Given that \(r\) can vary
b. use calculus to find the value of \(r\) for which \(S\) is stationary.
c. Find, to 3 significant figures the minimum value of \(S\).

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = x ( x + 2 ) ( x - 4 )\).
The region \(R _ { 1 }\) shown shaded in Figure 2 is bounded by the curve and the negative \(x\)-axis.

1. Show that the exact area of \(R _ { 1 }\) is \(\frac { 20 } { 3 }\) The region \(R _ { 2 }\) also shown shaded in Figure 2 is bounded by the curve, the positive \(x\)-axis and the line with equation \(x = b\), where \(b\) is a positive constant and \(0 < b < 4\) Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\)
2. verify that \(b\) satisfies the equation $$( b + 2 ) ^ { 2 } \left( 3 b ^ { 2 } - 20 b + 20 \right) = 0$$ The roots of the equation \(3 b ^ { 2 } - 20 b + 20 = 0\) are 1.225 and 5.442 to 3 decimal places. The value of \(b\) is therefore 1.225 to 3 decimal places.
3. Explain, with the aid of a diagram, the significance of the root 5.442

15. \begin{figure}[h] \end{figure} A company makes toys for children.
Figure 5 shows the design for a solid toy that looks like a piece of cheese.
The toy is modelled so that

• face \(A B C\) is a sector of a circle with radius \(r \mathrm {~cm}\) and centre \(A\)
• angle \(B A C = 0.8\) radians
• faces \(A B C\) and \(D E F\) are congruent
• edges \(A D , C F\) and \(B E\) are perpendicular to faces \(A B C\) and \(D E F\)
• edges \(A D , C F\) and \(B E\) have length \(h \mathrm {~cm}\)

Given that the volume of the toy is \(240 \mathrm {~cm} ^ { 3 }\)

1. show that the surface area of the toy, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 0.8 r ^ { 2 } + \frac { 1680 } { r }$$ making your method clear. Using algebraic differentiation,
2. find the value of \(r\) for which \(S\) has a stationary point.
3. Prove, by further differentiation, that this value of \(r\) gives the minimum surface area of the toy.

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of a curve with equation $$y = \frac { ( x - 2 ) ( x - 4 ) } { 4 \sqrt { x } } \quad x > 0$$ The region \(R\), shown shaded in Figure 3, is bounded by the curve and the \(x\)-axis.
Find the exact area of \(R\), writing your answer in the form \(a \sqrt { 2 } + b\), where \(a\) and \(b\) are constants to be found.