Review practice · Section 2 of 3

(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 marks

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

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

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

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

(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 marks

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.

The curve $C$ is $y=x\cos2x$, $0\le x\le\pi$.

(i) Find its exact x-axis intercepts.3 marks

(ii) Sketch $C$, stating all x- and y-axis intercept coordinates.2 marks

(iii) Find the exact value of $\int_{\pi/4}^\pi|x\cos2x|\,dx$.4 marks

(iv) Find the volume when the region bounded by $C$, the x-axis and $x=\pi$ rotates completely about the x-axis.2 marks

Sketch $y=e^{\sqrt x}$, $x\ge0$. 1 mark Rotate the region bounded by this curve, $x=4$ and both axes completely about the x-axis. Use $t=\sqrt x$ to show the volume is $2\pi\int_0^2te^{2t}\,dt$, and evaluate it exactly.6 marks

The diagram shows $y^2=6-x$ and $y=-x$.

(i) State their intersection coordinates.

(ii) Find the exact area of the finite region between the two graphs.

(iii) The shaded region $R$ is bounded by the y-axis, the line and the lower part of the parabola. Find the volume when $R$ rotates through $2\pi$ about the y-axis, to 2 decimal places.