Volume of revolution with parts

A question is this type if and only if it requires finding a volume of revolution where the integral for the volume necessitates integration by parts.

3 questions · Standard +0.5

1.08i Integration by parts 4.08d Volumes of revolution: about x and y axes

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AQA C3 2012 June Q4

7 marks Standard +0.3

By using integration by parts, find $\int x \mathrm { e } ^ { 6 x } \mathrm {~d} x$.
(4 marks)
The diagram shows part of the curve with equation $y = \sqrt { x } \mathrm { e } ^ { 3 x }$. \includegraphics[max width=\textwidth, alt={}, center]{d3c66c34-b09c-4223-8383-cf0a68419bf9-4_547_846_536_591} The shaded region $R$ is bounded by the curve $y = \sqrt { x } \mathrm { e } ^ { 3 x }$, the line $x = 1$ and the $x$-axis from $x = 0$ to $x = 1$. Find the volume of the solid generated when the region $R$ is rotated through $360 ^ { \circ }$ about the $x$-axis, giving your answer in the form $\pi \left( p \mathrm { e } ^ { 6 } + q \right)$, where $p$ and $q$ are rational numbers.
(3 marks)

AQA C3 2014 June Q6

9 marks Standard +0.8

By using integration by parts twice, find $$\int x ^ { 2 } \sin 2 x d x$$
A curve has equation $y = x \sqrt { \sin 2 x }$, for $0 \leqslant x \leqslant \frac { \pi } { 2 }$. The region bounded by the curve and the $x$-axis is rotated through $2 \pi$ radians about the $x$-axis to generate a solid. Find the exact value of the volume of the solid generated.
[0pt] [3 marks]

AQA C3 2011 June Q9

11 marks Standard +0.3

Use integration by parts to find $\int x\ln x \, dx$. [3]
Given that $y = (\ln x)^2$, find $\frac{dy}{dx}$. [2]
The diagram shows part of the curve with equation $y = \sqrt{x\ln x}$. \includegraphics{figure_9} The shaded region $R$ is bounded by the curve $y = \sqrt{x\ln x}$, the line $x = e$ and the $x$-axis from $x = 1$ to $x = e$. Find the volume of the solid generated when the region $R$ is rotated through 360° about the $x$-axis, giving your answer in an exact form. [6]