Practice section 7 of 9

Find $\displaystyle\int_0^\pi x^2\sin(nx)\,dx$, where $n$ is an even integer. Leave the answer in the form $k\pi^2/n$ and determine $k$. [4]

(a) The ellipse $x^2/a^2+y^2/b^2=1$ has $b>a>0$.

(i) Show its enclosed area is $A=\tfrac{4b}{a}\int_0^a\sqrt{a^2-x^2}\,dx$. [1] Use $x=a\sin\theta$ to find $A$ in terms of $a,b,\pi$. [4]

(ii) Rotate the enclosed region through $\pi$ radians about the x-axis to form an ellipsoid. Find its volume. [3]

(b) The curved surface area generated by completely rotating a nonnegative curve $y=f(x)$, $a\le x\le b$, about the x-axis is

Use the circle $x^2+y^2=r^2$, $-r\le x\le r$, to show the surface area of a sphere is $4\pi r^2$. [4]

(a) Find the exact value of $\int_1^2x^2\ln x\,dx$. [3]

(b) Find $\int9/\sqrt{3+2x-x^2}\,dx$. Hence find the exact constant $a$ satisfying

[7]

(a)(i) For $a>1$, use $u=1/x$ to show $\int_1^a dx/(1+x^2)=\int_{1/a}^1du/(1+u^2)$. [4]

(ii) Hence show $\tan^{-1}a+\tan^{-1}(1/a)=k$, finding $k$. [3]

(b)(i) Show $\int_0^1x^3/(1+x^2)\,dx=1/2-(1/2)\ln2$. [2]

(ii) Without using a calculator, evaluate $\int_0^1x^2\tan^{-1}x\,dx$. [3]

(a) Find $\int\sin^{-1}(3x)\,dx$. [3]

(b) The curves $C_1,C_2$ are $y=1-(\sqrt3/\pi)x^2$ and $y=\tfrac13\cos x$. Rotate the first-quadrant region bounded by them and the y-axis through $360^\circ$ about the y-axis. Find the volume to 2 decimal places. [4]

The diagram shows $y=\sin^{-1}(x/2)$ for $0\le x\le2$.

(i) Find the area of $R_1$, bounded by the curve, $y=\pi/12$, $y=\pi/6$ and the y-axis.

(ii) Find the volume when $R_1$ rotates through $2\pi$ about the x-axis.

(iii) Without using a calculator, find the exact area of $R_2$, bounded by the curve, $x=1$, $x=\sqrt3$ and the x-axis.