Let \(I_{n}=\int_{0}^{1} x^{n}(1-x)^{\frac{1}{2}} \mathrm{~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 1\),
\((3+2 n) I_{n}=2 n I_{n-1} .\)
Hence find the exact value of \(I_{3}\).
Show that \(\frac{\mathrm{d}}{\mathrm{d} t}\left(t\left(1+t^{3}\right)^{n}\right)=(3 n+1)\left(1+t^{3}\right)^{n}-3 n\left(1+t^{3}\right)^{n-1}\).
Let \(I_{n}=\int_{0}^{1}\left(1+t^{3}\right)^{n} \mathrm{~d} t\). Using the above result, or otherwise, show that
\((3 n+1) I_{n}=2^{n}+3 n I_{n-1}\)
Hence evaluate \(I_{3}\).
Let
\(I_{n}=\int_{0}^{1} t^{n} \mathrm{e}^{-t} \mathrm{~d} t\)
where \(n \geqslant 0\). Show that, for all \(n \geqslant 1\),
\(I_{n}=n I_{n-1}-\mathrm{e}^{-1}\)
Hence prove by induction that, for all positive integers \(n\),
\(I_{n}\lt n!.\)
Show that
\(\frac{\mathrm{d}}{\mathrm{~d} x}\left[x^{n-1} \sqrt{ }\left(4-x^{2}\right)\right]=\frac{4(n-1) x^{n-2}}{\sqrt{ }\left(4-x^{2}\right)}-\frac{n x^{n}}{\sqrt{ }\left(4-x^{2}\right)} .\)
Let
\(I_{n}=\int_{0}^{1} \frac{x^{n}}{\sqrt{ }\left(4-x^{2}\right)} \mathrm{d} x\)
where \(n \geqslant 0\). Prove that
\(n I_{n}=4(n-1) I_{n-2}-\sqrt{ } 3,\)
for \(n \geqslant 2\).
Given that \(I_{0}=\frac{1}{6} \pi\), find \(I_{4}\), leaving your answer in an exact form.
5 A curve \(C\) is defined parametrically by
\(x=\frac{2}{\mathrm{e}^{t}+\mathrm{e}^{-t}} \quad \text { and } \quad y=\frac{\mathrm{e}^{t}-\mathrm{e}^{-t}}{\mathrm{e}^{t}+\mathrm{e}^{-t}},\)
for \(0 \leqslant t \leqslant 1\). The area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(S\).
(i) Show that \(S=4 \pi \int_{0}^{1} \frac{\mathrm{e}^{t}-\mathrm{e}^{-t}}{\left(\mathrm{e}^{t}+\mathrm{e}^{-t}\right)^{2}} \mathrm{~d} t\).
(ii) Using the substitution \(u=\mathrm{e}^{t}+\mathrm{e}^{-t}\), or otherwise, find \(S\) in terms of \(\pi\) and e .
The curve \(C\) is defined parametrically by
\(x=\mathrm{e}^{t}-t, \quad y=4 \mathrm{e}^{\frac{1}{2} t} .\)
Find the length of the arc of \(C\) from the point where \(t=0\) to the point where \(t=3\).
A curve is defined parametrically by
\(x=t-\frac{1}{2} \sin 2 t \quad \text { and } \quad y=\sin ^{2} t\)
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 \(S\).
(i) Show that
\(S=a \pi \int_{0}^{\pi} \sin ^{3} t \mathrm{~d} t\)
where the constant \(a\) is to be found.
(ii) Using the result \(\sin 3 t=3 \sin t-4 \sin ^{3} t\), find the exact value of \(S\).
A curve has equation \(y=\mathrm{e}^{x}\) for \(\ln \frac{4}{3} \leqslant x \leqslant \ln \frac{12}{5}\). The area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(A\).
(a) Use the substitution \(u=\mathrm{e}^{x}\) to show that
\(A=2 \pi \int_{\frac{4}{3}}^{\frac{12}{5}} \sqrt{1+u^{2}} \mathrm{~d} u\)
(b) Use the substitution \(u=\sinh v\) to show that
\(A=\pi\left(\frac{904}{225}+\ln \frac{5}{3}\right)\)
(a) Starting from the definitions of cosh and sinh in terms of exponentials, prove that
\(2 \sinh ^{2} A=\cosh 2 A-1\)
(b) A curve has equation \(y=x^{2}\), for \(0 \leqslant x \leqslant \frac{2}{3}\). The area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(S\).
Use the substitution \(x=\frac{1}{2} \sinh u\) to show that \(S=\frac{1}{32} \pi\left(\frac{820}{81}-\ln 3\right)\).
(a) A curve has equation \(y=\mathrm{e}^{x}+\frac{1}{4} \mathrm{e}^{-x}\), for \(0 \leqslant x \leqslant 1\). Find, in terms of \(\pi\) and e , the area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
(b) Using standard results from the list of formulae (MF19), or otherwise, find the Maclaurin's series for \(\mathrm{e}^{x}+\frac{1}{4} \mathrm{e}^{-x}\) up to and including the term in \(x^{2}\).
The curve \(C\) has parametric equations
\(x=2 \cosh t, \quad y=\frac{3}{2} t-\frac{1}{4} \sinh 2 t, \text { for } 0 \leqslant t \leqslant 1 .\)
(a) Find \(\frac{\mathrm{d} x}{\mathrm{~d} t}\) and show that \(\frac{\mathrm{d} y}{\mathrm{~d} t}=1-\sinh ^{2} t\).
The area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(A\).
(b) (i) Show that \(A=\pi \int_{0}^{1}\left(\frac{3}{2} t-\frac{1}{4} \sinh 2 t\right)(1+\cosh 2 t) \mathrm{d} t\).
(ii) Hence find \(A\) in terms of \(\pi, \sinh 2\) and \(\cosh 2\).
(a) Starting from the definition of cosh in terms of exponentials, prove that
\(2 \cosh ^{2} A=\cosh 2 A+1\)
The curve \(C\) has parametric equations
\(x=2 \cosh 2 t+3 t, \quad y=\frac{3}{2} \cosh 2 t-4 t, \quad \text { for }-\frac{1}{2} \leqslant t \leqslant \frac{1}{2} .\)
The area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(y\)-axis is denoted by \(A\).
(b) (i) Show that \(A=10 \pi \int_{-\frac{1}{2}}^{\frac{1}{2}}(2 \cosh 2 t+3 t) \cosh 2 t \mathrm{~d} t\).
(ii) Hence find \(A\) in terms of \(\pi\) and e.
The curves \(C_{1}: y=\cosh x\) and \(C_{2}: y=\sinh 2 x\) intersect at the point where \(x=a\).
(a) Find the exact value of \(a\), giving your answer in logarithmic form.
(b) Sketch \(C_{1}\) and \(C_{2}\) on the same diagram.
(c) Find the exact value of the length of the \(\operatorname{arc}\) of \(C_{1}\) from \(x=0\) to \(x=a\).
A curve has equation \(y=\cosh x\), for \(0 \leqslant x \leqslant \frac{1}{2}\).
Find, in terms of \(\pi\) and e , the area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
The curve \(C\) is defined parametrically by \(x=18t-t^2\) and \(y=8t^{3/2}\), where \(0\lt t\lt4\).
(i) Show that, at all points of \(C\), \(\frac{d^2y}{dx^2}=\frac{3(9+t)}{2t^{1/2}(9-t)^3}\).
(ii) Show that the mean value of \(\frac{d^2y}{dx^2}\) with respect to \(x\) over the interval \(0\le x\le56\) is \(\frac{3}{70}\).
(iii) Find the area of the surface generated when \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis, showing full working.
The curve \(C\) has polar equation \(r=a(1+\sin \theta)\) for \(-\pi\lt \theta \leqslant \pi\), where \(a\) is a positive constant.
(i) Sketch \(C\).
(ii) Find the area of the region enclosed by \(C\).
(iii) Show that the length of the arc of \(C\) from the pole to the point furthest from the pole is given by
\(s=(\sqrt{ } 2) a \int_{-\frac{1}{2} \pi}^{\frac{1}{2} \pi} \sqrt{ }(1+\sin \theta) \mathrm{d} \theta\).
(iv) Show that the substitution \(u=1+\sin \theta\) reduces this integral for \(s\) to \((\sqrt{ } 2) a \int_{0}^{2} \frac{1}{\sqrt{ }(2-u)} \mathrm{d} u\). Hence evaluate \(s\).
The curve \(C\) has equation \(y=\frac12(e^x+e^{-x})\), for \(0\le x\le4\).
(i) The region \(R\) is bounded by \(C\), the \(x\)-axis, the \(y\)-axis and the line \(x=4\). Find, in terms of \(e\), the coordinates of the centroid of the region \(R\).
(ii) Show that \(\frac{ds}{dx}=\frac12(e^x+e^{-x})\), where \(s\) denotes the arc length of \(C\), and find the surface area generated when \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis.
A curve \(C\) has parametric equations
\(x=\frac{2}{5} t^{\frac{5}{2}}-2 t^{\frac{1}{2}}, \quad y=\frac{4}{3} t^{\frac{3}{2}}, \quad \text { for } 1 \leqslant t \leqslant 4\)
(i) Find the exact value of the arc length of \(C\).
(ii) Find also the exact value of the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
The curve \(C\) has parametric equations
\(x=\mathrm{e}^{t}-4 t+3, \quad y=8 \mathrm{e}^{\frac{1}{2} t}, \quad \text { for } 0 \leqslant t \leqslant 2 .\)
(i) Find, in terms of e , the length of \(C\).
(ii) Find, in terms of \(\pi\) and e , the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
The curve \(C\) has parametric equations
\(x=t^{2}, \quad y=t-\frac{1}{3} t^{3}, \quad \text { for } 0 \leqslant t \leqslant 1 .\)
Find
(i) the arc length of \(C\),
(ii) the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.