Practice section 9 of 9

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]

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

(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]

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]

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

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

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]

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

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