Review practice · Section 3 of 3

(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 $a$.

Designer A programs a machine to cut a thin material into the region enclosed by $y=\sqrt{2x-3}$, $2y=x-4$ and the x-axis.

(i) Sketch suitable graphs and find its exact surface area.5 marks

(ii) Rotate that region through four right angles about the x-axis to form a closed object. Find its volume.3 marks

(iii) Designer B rotates the region bounded by $C:y=x^2+3$, $x=4$ and both axes through four right angles about $x=4$. Write the equation of $C$ translated four units in the negative x-direction, and determine the resulting volume.4 marks

A force field is $\mathbf F=M(x,y)\mathbf i+N(x,y)\mathbf j$. For a path $y=f(x)$ from $x=a$ to $x=b$, the work is

$$W=\int_a^b\left[M(x,y)+N(x,y)\frac{dy}{dx}\right]dx.$$

A field is conservative if the work from one point to another is independent of the path.

(i) Sketch $C:y^2=4(1-x)$, $x\le1$.2 marks

(ii) Find $dy/dx$ in terms of $y$.1 mark

(iii) Points $P,Q$ on $C$ have $x=1,-3$ respectively, with $Q$ below the x-axis. Find line $PQ$.2 marks

For the remaining parts, $\mathbf F=x^2\mathbf i+xy^2\mathbf j$.

(iv) Show that work along $C$ from $Q$ to $P$ is $\int_{-3}^1[x^2+4x(1-x)^{1/2}]dx$, then evaluate it exactly.4 marks

(v) Find the work along line $PQ$ from $Q$ to $P$ to 2 decimal places.2 marks

(vi) Determine whether $\mathbf F$ is conservative, with a reason.1 mark