Practice section 8 of 9

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

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

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

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

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

Sketch $y=e^{\sqrt x}$, $x\ge0$. [1] 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]

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.

(i) Find $\int\ln(2x-1)\,dx$.

(ii) Sketch $y=|\ln(2x-1)|$, stating asymptotes and axis intercepts.

(iii) Find the exact area bounded by $y=|\ln(2x-1)|$, the x-axis, $x=3/4$ and $x=3/2$.

(iv) Find the volume when the region bounded by $y=\ln(2x-1)$, the x-axis, $x=3/2$ and $x=3$ rotates completely about the y-axis.

Let $f(x)=\ln x$ for $1\le x\le e$, and $f(x)=0$ for $e<x\le3$, with $f(x+2)=f(x)$ for all real $x$.

(a) Rotate the region bounded by $y=f(x)$, the x-axis and $x=e$ through $2\pi$ about the x-axis. Find the exact volume.

(b) Sketch $y=f(x)$ for $-3\le x\le4$. Hence find $\int_{-3}^4f(x)dx$.

Define $f(x)=4/[(2x-5)^2+1]$ for real $x\le5/2$.

(i) Sketch $y=f(x)$ and show $f^{-1}$ exists.

(ii) Find $f^{-1}(x)$ and its domain.

The intersections of $f$ and $f^{-1}$ have x-coordinates $a$ and $2$, where $a<2$. Find the enclosed area in terms of $a$.