Practice section 6 of 9

Use $x=\sqrt{12}\cos\theta$ to show that $\displaystyle\int_0^{\sqrt3}\sqrt{12-x^2}\,dx=\pi+3\sqrt3/2$. Region $A$ is bounded by $y=x^2$ and a minor arc of $x^2+y^2=12$.

(i) Find its exact area.

(ii) Find the exact volume when $A$ rotates through $\pi$ radians about the y-axis.

Region $R$ is bounded by $y=2x$, $y=3/2$, the x-axis and $y=\sqrt{(3x^2-1)/x^2}$, as shown. Find the volume when $R$ rotates through $2\pi$ about the x-axis, correct to three decimal places.

Curve $C$ is $y=\sqrt{3x/(4-x)}$, $0\le x<4$. Region $R$ is bounded by $C$, $y=3$ and the y-axis. It rotates through $2\pi$ about $y=3$. Write the equation after translating $C$ three units down, and hence find the solid volume.

The region $R$ is bounded by the x-axis, y-axis, $y=1$ and $y=\ln x$, $x>0$. The diagram approximates its area by $A$, the total area of $n$ rectangles of height $1/n$, whose widths use the lower y-endpoint of each strip. Show that

Another finite region $S$ is bounded by the x-axis, $x=e$ and $y=\ln x$, $x>0$. Explain how $A$ approximates the area of $S$, and whether this underestimates or overestimates it, giving a reason. Find the exact volume when $S$ rotates completely about the y-axis.

(i) For continuous $f$, explain with a sketch why

[2]

(ii) Hence evaluate $\displaystyle\lim_{n\to\infty}\frac1n\left(\frac{\sqrt[3]1+\sqrt[3]2+\cdots+\sqrt[3]n}{\sqrt[3]n}\right)$. [3]

Find $\displaystyle\int_0^a x\sqrt{1+6x^2}\,dx$.