1.08d Evaluate definite integrals: between limits

\includegraphics{figure_5} The diagram shows part of the curve \(x = \frac{8}{y^2} - 2\), crossing the \(y\)-axis at the point \(A\). The point \(B (6, 1)\) lies on the curve. The shaded region is bounded by the curve, the \(y\)-axis and the line \(y = 1\). Find the exact volume obtained when this shaded region is rotated through \(360°\) about the \(y\)-axis. [6]

\includegraphics{figure_4} The diagram shows the curve \(y = e^x + 4e^{-2x}\) and its minimum point \(M\).

1. Show that the \(x\)-coordinate of \(M\) is \(\ln 2\). [3]
2. The region shaded in the diagram is enclosed by the curve and the lines \(x = 0\), \(x = \ln 2\) and \(y = 0\). Use integration to show that the area of the shaded region is \(\frac{5}{2}\). [4]

\includegraphics{figure_6} The diagram shows part of the curve with equation $$y = 4\sin^2 x + 8\sin x + 3$$ and its point of intersection \(P\) with the \(x\)-axis.

1. Find the exact \(x\)-coordinate of \(P\). [3]
2. Show that the equation of the curve can be written $$y = 5 + 8\sin x - 2\cos 2x,$$ and use integration to find the exact area of the shaded region enclosed by the curve and the axes. [6]

1. By differentiating \(\frac{\cos x}{\sin x}\), show that if \(y = \cot x\) then \(\frac{dy}{dx} = -\cosec^2 x\). [3]
2. Hence show that \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cosec^2 x \, dx = \sqrt{3}\). [2]

By using appropriate trigonometrical identities, find the exact value of

1. \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cot^2 x \, dx\), [3]
2. \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{1}{1 - \cos 2x} \, dx\). [3]

It is given that the positive constant \(a\) is such that $$\int_{-a}^a (4e^{2x} + 5) dx = 100.$$

1. Show that \(a = \frac{1}{4}\ln(50 + e^{-2a} - 5a)\). [4]
2. Use the iterative formula \(a_{n+1} = \frac{1}{4}\ln(50 + e^{-2a_n} - 5a_n)\) to find \(a\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places. [3]

\includegraphics{figure_5} The diagram shows the curve \(y = \sqrt{1 + e^{4x}}\) for \(0 \leq x \leq 6\). The region bounded by the curve and the lines \(x = 0\), \(x = 6\) and \(y = 0\) is denoted by \(R\).

1. Use the trapezium rule with 2 strips to find an estimate of the area of \(R\), giving your answer correct to 2 decimal places. [3]
2. With reference to the diagram, explain why this estimate is greater than the exact area of \(R\). [1]
3. The region \(R\) is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced. [4]

Show that \(\int_1^7 \frac{6}{2x + 1} \, dx = \ln 125\). [5]

Let \(\text{f}(x) = \frac{4x^2 + 9x - 8}{(x + 2)(2x - 1)}\).

1. Express \(\text{f}(x)\) in the form \(A + \frac{B}{x + 2} + \frac{C}{2x - 1}\). [4]
2. Hence show that \(\int_1^4 \text{f}(x) \, dx = 6 + \frac{1}{2} \ln\left(\frac{16}{7}\right)\). [5]

A curve is defined parametrically by $$x = t - \frac{1}{2}\sin 2t \quad \text{and} \quad y = \sin^2 t.$$ The arc of the curve joining the point where \(t = 0\) to the point where \(t = \pi\) is rotated through one complete revolution about the \(x\)-axis. The area of the surface generated is denoted by \(S\).

1. Show that $$S = a\pi \int_0^\pi \sin^3 t \, dt,$$ where the constant \(a\) is to be found. [5]
2. Using the result \(\sin 3t = 3\sin t - 4\sin^3 t\), find the exact value of \(S\). [3]

1. Use de Moivre's theorem to show that \(\sin^6 \theta = -\frac{1}{32}(\cos 6\theta - 6\cos 4\theta + 15\cos 2\theta - 10)\). [6]

It is given that \(\cos^6 \theta = \frac{1}{32}(\cos 6\theta + 6\cos 4\theta + 15\cos 2\theta + 10)\).

1. Find the exact value of \(\int_0^{\frac{1}{4}\pi}\left(\cos^6\left(\frac{1}{4}x\right) + \sin^6\left(\frac{1}{4}x\right)\right)dx\). [4]
2. Express each root of the equation \(16c^6 + 16\left(1-c^2\right)^3 - 13 = 0\) in the form \(\cos k\pi\), where \(k\) is a rational number. [5]

A particle moves in a straight line \(AB\). The velocity \(v\text{ m s}^{-1}\) of the particle \(t\text{ s}\) after leaving \(A\) is given by \(v = t(5 - 2t)\) where \(k\) is a constant. The displacement of the particle from \(A\), in the direction towards \(B\), is \(2.5\text{ m}\) when \(t = 3\) and is \(2.4\text{ m}\) when \(t = 6\).

1. Find the value of \(k\). Hence find an expression, in terms of \(t\), for the displacement of the particle from \(A\). [7]
2. Find the displacement of the particle from \(A\) when its velocity is a minimum. [4]

A particle moves in a straight line starting from a point \(O\). The velocity \(v\) m s\(^{-1}\) of the particle \(t\) s after leaving \(O\) is given by $$v = t^3 - \frac{9}{2}t^2 + 1 \text{ for } 0 \leqslant t \leqslant 4.$$ You may assume that the velocity of the particle is positive for \(t < \frac{1}{2}\), is zero at \(t = \frac{1}{2}\) and is negative for \(t > \frac{1}{2}\).

1. Find the distance travelled between \(t = 0\) and \(t = \frac{1}{2}\). [4]
2. Find the positive value of \(t\) at which the acceleration is zero. Hence find the total distance travelled between \(t = 0\) and this instant. [4]

A particle \(P\) moves in a straight line. It starts from rest at a point \(O\) on the line and at time \(t\) s after leaving \(O\) it has acceleration \(a \text{ m s}^{-2}\), where \(a = 6t - 18\). Find the distance \(P\) moves before it comes to instantaneous rest. [6]

A particle \(P\) moves on the \(x\)-axis from the origin \(O\) with an initial velocity of \(-20\) m s\(^{-1}\). The acceleration \(a\) m s\(^{-2}\) at time \(t\) s after leaving \(O\) is given by \(a = 12 - 2t\).

1. Sketch a velocity-time graph for \(0 \leq t \leq 12\), indicating the times when \(P\) is at rest. [5]
2. Find the total distance travelled by \(P\) in the interval \(0 \leq t \leq 12\). [5]

A car of mass 1200 kg is travelling along a straight horizontal road \(AB\). There is a constant resistance force of magnitude 500 N. When the car passes point \(A\), it has a speed of \(15 \text{ m s}^{-1}\) and an acceleration of \(0.8 \text{ m s}^{-2}\).

1. Find the power of the car's engine at the point \(A\). [3]

The car continues to work with this power as it travels from \(A\) to \(B\). The car takes 53 seconds to travel from \(A\) to \(B\) and the speed of the car at \(B\) is \(32 \text{ m s}^{-1}\).

1. Show that the distance \(AB\) is 1362.6 m. [3]

A vehicle is moving in a straight line. The velocity \(v \text{ m s}^{-1}\) at time \(t \text{ s}\) after the vehicle starts is given by $$v = A(t - 0.05t^2) \text{ for } 0 \leq t \leq 15,$$ $$v = \frac{B}{t} \text{ for } t \geq 15,$$ where \(A\) and \(B\) are constants. The distance travelled by the vehicle between \(t = 0\) and \(t = 15\) is \(225 \text{ m}\).

1. Find the value of \(A\) and show that \(B = 3375\). [5]
2. Find an expression in terms of \(t\) for the total distance travelled by the vehicle when \(t \geq 15\). [3]
3. Find the speed of the vehicle when it has travelled a total distance of \(315 \text{ m}\). [3]

A particle of mass \(0.5\) kg moves in a straight line under the action of a variable force. At time \(t\) seconds, the force is \((3t - 2)\) N in the direction of motion. Given that the particle starts from rest, find the velocity of the particle when \(t = 4\). [5]

A particle \(P\) moves on a straight line. It starts at a point \(O\) on the line and returns to \(O\) 100 s later. The velocity of \(P\) is \(v \text{ m s}^{-1}\) at time \(t\) s after leaving \(O\), where $$v = 0.0001t^3 - 0.015t^2 + 0.5t.$$

1. Show that \(P\) is instantaneously at rest when \(t = 0\), \(t = 50\) and \(t = 100\). [2]
2. Find the values of \(v\) at the times for which the acceleration of \(P\) is zero, and sketch the velocity-time graph for \(P\)'s motion for \(0 \leq t \leq 100\). [7]
3. Find the greatest distance of \(P\) from \(O\) for \(0 \leq t \leq 100\). [4]

A particle \(P\) moves in a straight line starting from a point \(O\). The velocity \(v\text{ m s}^{-1}\) of \(P\) at time \(t\text{ s}\) is given by $$v = 12t - 4t^2 \quad \text{for } 0 \leqslant t \leqslant 2,$$ $$v = 16 - 4t \quad \text{for } 2 \leqslant t \leqslant 4.$$

1. Find the maximum velocity of \(P\) during the first \(2\text{ s}\). [3]
2. Determine, with justification, whether there is any instantaneous change in the acceleration of \(P\) when \(t = 2\). [2]
3. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 4\). [3]
4. Find the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\). [5]

Particles \(P\) and \(Q\) leave a fixed point \(A\) at the same time and travel in the same straight line. The velocity of \(P\) after \(t\) seconds is \(6(t - 3)\) m s\(^{-1}\) and the velocity of \(Q\) after \(t\) seconds is \((10 - 2t)\) m s\(^{-1}\).

1. Sketch, on the same axes, velocity-time graphs for \(P\) and \(Q\) for \(0 \leq t \leq 5\). [3]
2. Verify that \(P\) and \(Q\) meet after 5 seconds. [4]
3. Find the greatest distance between \(P\) and \(Q\) for \(0 \leq t \leq 5\). [4]

A particle \(P\) moves in a straight line starting from a point \(O\) and comes to rest \(35\) s later. At time \(t\) s after leaving \(O\), the velocity \(v\) m s\(^{-1}\) of \(P\) is given by $$v = \frac{4}{5}t^2 \quad 0 \leq t \leq 5,$$ $$v = 2t + 10 \quad 5 \leq t \leq 15,$$ $$v = a + bt^2 \quad 15 \leq t \leq 35,$$ where \(a\) and \(b\) are constants such that \(a > 0\) and \(b < 0\).

1. Show that the values of \(a\) and \(b\) are \(49\) and \(-0.04\) respectively. [3]
2. Sketch the velocity-time graph. [4]
3. Find the total distance travelled by \(P\) during the \(35\) s. [5]