1.08d Evaluate definite integrals: between limits

The function \(f\) is defined by \(f(x) = \frac{4x^2 + 12x + 9}{2x^2 + x - 3}\), where \(x > 1\).

1. Show that \(f(x)\) can be written as \(2 + \frac{5}{x-1}\). [3]
2. Hence find the exact value of \(\int_3^7 f(x)\,dx\). [4]

The function \(f\) is given by $$f(x) = \frac{25x + 32}{(2x - 5)(x + 1)(x + 2)}.$$

1. Express \(f(x)\) in terms of partial fractions. [4]
2. Show that \(\int_1^2 f(x) dx = -\ln P\), where \(P\) is an integer whose value is to be found. [5]
3. Show that the sign of \(f(x)\) changes in the interval \(x = 2\) to \(x = 3\). Explain why the change of sign method fails to locate a root of the equation \(f(x) = 0\) in this case. [2]

Let \(a, b\) satisfy \(0 < a < b\).

1. Find, in terms of \(a\) and \(b\), the value of $$\int_a^b \frac{81}{x^4} dx$$ [2]
2. Explaining clearly any limiting processes used, find the value of \(a\), given that $$\int_a^{\infty} \frac{81}{x^4} dx = \frac{216}{125}$$ [2]

The diagram below shows part of a curve C with equation \(y = 1 + 3x - \frac{1}{2}x^2\). \includegraphics{figure_7}

1. The curve crosses the \(y\) axis at the point A. The straight line L is normal to the curve at A and meets the curve again at B. Find the equation of L and the \(x\) coordinate of the point B. [4]
2. The region R is bounded by the curve C and the line L. Find the exact area of R. [3]

This is the graph of \(y = \frac{5}{4x-3} - \frac{3}{2}\) \includegraphics{figure_7} Find the area between the graph, the \(x\) axis, and the lines \(x = 1\) and \(x = 7\) in the form \(a \ln b + c\) where \(a, b, c \in Q\) [6]

This is the graph of \(y = \frac{5}{4x-3} - \frac{3}{2}\) \includegraphics{figure_6} Find the area between the graph, the \(x\) axis, and the lines \(x = 1\) and \(x = 7\) in the form \(a \ln b + c\) where \(a,b,c \in Q\) [6]

Given that \(k\) is a positive constant and \(\int_1^k \left(\frac{5}{2\sqrt{x}} + 3\right)dx = 4\)

1. show that \(3k + 5\sqrt{k} - 12 = 0\) [4]
2. Hence, using algebra, find any values of \(k\) such that $$\int_1^k \left(\frac{5}{2\sqrt{x}} + 3\right)dx = 4$$ [4]

\includegraphics{figure_5} Figure 5 shows a sketch of the curve with parametric equations $$x = 3\cos 2t, \quad y = 2\tan t, \quad 0 \leq t \leq \frac{\pi}{4}.$$ The region \(R\), shown shaded in Figure 5, is bounded by the curve, the \(x\)-axis and the \(y\)-axis.

1. Show that the area of \(R\) is given $$\int_0^{\pi/4} 24\sin^2 t \, dt.$$ [4]
2. Hence, using algebraic integration, find the exact area of \(R\). [3]

\includegraphics{figure_4} Figure 4 shows a sketch of part of the curve with equation $$y = 2e^{2x} - xe^{2x}, \quad x \in \mathbb{R}$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the \(y\)-axis. Use calculus to show that the exact area of \(R\) can be written in the form \(pe^t + q\), where \(p\) and \(q\) are rational constants to be found. (Solutions based entirely on graphical or numerical methods are not acceptable.) [5]

\includegraphics{figure_3} Figure 3 shows a sketch of the curve with equation \(y = 2x^2 - x\). The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the line with equation \(x = -0.5\), the \(x\)-axis and the line with equation \(x = 1.5\).

1. The trapezium rule with four strips is used to find an estimate for the area of \(R\). Explain whether the estimate for R is an underestimate or overestimate to the true value for the area of \(R\). [1]

The estimate for R is found to be 2.58. Using this value, and showing your working,

1. estimate the value of \(\int_{-0.5}^{1.5} (2x^2 + 1 + 2x) \, dx\). [3]

Find the value of the integral: $$\int_0^1 \frac{x^{\frac{1}{2}} + x^{-\frac{1}{3}}}{x} \, dx$$ [4]

\includegraphics{figure_9} A mathematics student is modelling the profile of a glass bottle of water. Figure 1 shows a sketch of a central vertical cross-section \(ABCDEFGHA\) of the bottle with measurements taken by the student. The horizontal cross-section between \(CF\) and \(DE\) is a circle of diameter 8 cm and the horizontal cross-section between \(BG\) and \(AH\) is a circle of diameter 2 cm. The student thinks that the curve \(GF\) could be modelled as a curve with equation $$y = ax^2 + b \qquad 1 \leq x \leq 4$$ where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.

1. Find the value of \(a\) and the value of \(b\) according to the model. [2]
2. Use the model to find the volume of water that the bottle can contain. [7]

\includegraphics{figure_9} Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x \ln x, \quad x > 0\) The line \(l\) is the normal to \(C\) at the point \(P(e, e)\) The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(x\)-axis. Show that the exact area of \(R\) is \(Ae^2 + B\) where \(A\) and \(B\) are rational numbers to be found. [9]

\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]

Let \(f(x) = \frac{1 + x^2}{\sqrt{4 - 3x}}\)

1. Obtain in ascending powers of \(x\) the first three terms in the expansion of \(\frac{1}{\sqrt{4 - 3x}}\) and state the values of \(x\) for which this expansion is valid. [5]
2. Hence obtain an approximation to \(f(x)\) in the form \(a + bx + cx^2\) where \(a\), \(b\) and \(c\) are constants. [2]
3. Use your approximation to estimate \(\int_0^{0.1} f(x) dx\). [2]

Find the exact value of $$\int_1^4 \left(10x^2 - 3x^2\right) dx.$$ [3]

Find the exact value of \(\int_0^1 (e^x - x) dx\). [4]

A particle travels along a straight line. Its velocity \(v\) m s\(^{-1}\) after \(t\) seconds is given by $$v = t^3 - 9t^2 + 20t$$ When \(t = 0\), the particle is at rest at \(P\).

1. Find the times, other than \(t = 0\), at which the particle is at rest. [2]
2. Find the displacement of the particle from \(P\) when \(t = 2\). [4]

A particle travels along a straight line. Its velocity \(v\) ms\(^{-1}\) after \(t\) seconds is given by $$v = t^3 - 9t^2 + 20t$$ When \(t = 0\), the particle is at rest at \(P\).

1. Find the times, other than \(t = 0\), at which the particle is at rest. [2]
2. Find the displacement of the particle from \(P\) when \(t = 2\). [4]

% \includegraphics{figure_2} - Shows a circular tower with center T at (0,1), a goat at point G attached to the base at O, with string along arc OA then tangent AG A circular tower stands in a large horizontal field of grass. A goat is attached to one end of a string and the other end of the string is attached to the fixed point \(O\) at the base of the tower. Taking the point \(O\) as the origin \((0, 0)\), the centre of the base of the tower is at the point \(T(0, 1)\). The radius of the base of the tower is 1. The string has length \(\pi\) and you may ignore the size of the goat. The curve \(C\) represents the edge of the region that the goat can reach as shown in Figure 2.

1. Write down the equation of \(C\) for \(y < 0\). [1] When the goat is at the point \(G(x, y)\), with \(x > 0\) and \(y > 0\), as shown in Figure 2, the string lies along \(OAG\) where \(OA\) is an arc of the circle with angle \(OTA = \theta\) radians and \(AG\) is a tangent to the circle at \(A\).
2. With the aid of a suitable diagram show that $$x = \sin \theta + (\pi - \theta) \cos \theta$$ $$y = 1 - \cos \theta + (\pi - \theta) \sin \theta$$ [5]
3. By considering \(\int y \frac{dx}{d\theta} d\theta\), show that the area between \(C\), the positive \(x\)-axis and the positive \(y\)-axis can be expressed in the form $$\int_0^{\pi} u \sin u \, du + \int_0^{\pi} u^2 \sin^2 u \, du + \int_0^{\pi} u \sin u \cos u \, du$$ [5]
4. Show that \(\int_0^{\pi} u^2 \sin^2 u \, du = \frac{\pi^3}{6} + \int_0^{\pi} u \sin u \cos u \, du\) [4]
5. Hence find the area of grass that can be reached by the goat. [8]

4 A particle \(P\) moves on a straight line, starting from rest at a point \(O\) of the line. The time after \(P\) starts to move is \(t \mathrm {~s}\), and the particle moves along the line with constant acceleration \(\frac { 1 } { 4 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it passes through a point \(A\) at time \(t = 8\). After passing through \(A\) the velocity of \(P\) is \(\frac { 1 } { 2 } t ^ { \frac { 2 } { 3 } } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the acceleration of \(P\) immediately after it passes through \(A\). Hence show that the acceleration of \(P\) decreases by \(\frac { 1 } { 12 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) as it passes through \(A\).
2. Find the distance moved by \(P\) from \(t = 0\) to \(t = 27\).