Review practice · Section 1 of 3

The region $R$ is bounded by the x-axis, y-axis, $y=1$ and $y=\ln x$, $x>0$. The diagram approximates its area by $A$, the total area of $n$ rectangles of height $1/n$, whose widths use the lower y-endpoint of each strip. Show that

Another finite region $S$ is bounded by the x-axis, $x=e$ and $y=\ln x$, $x>0$. Explain how $A$ approximates the area of $S$, and whether this underestimates or overestimates it, giving a reason. Find the exact volume when $S$ rotates completely about the y-axis.

(i) For continuous $f$, explain with a sketch why

(ii) Hence evaluate $\displaystyle\lim_{n\to\infty}\frac1n\left(\frac{\sqrt[3]1+\sqrt[3]2+\cdots+\sqrt[3]n}{\sqrt[3]n}\right)$.3 marks

Find $\displaystyle\int_0^a x\sqrt{1+6x^2}\,dx$.

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 marks

(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 mark Use $x=a\sin\theta$ to find $A$ in terms of $a,b,\pi$.4 marks

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

(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 marks

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

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