5.17 Areas and volumes of revolution questions for IB Maths AA

Write $x^2+4x+13$ in the form $(x+p{)}^2+q$, where $p,q\in \mathbb{Z}$.

Hence find ${\displaystyle \int \frac{1}{x^2+4x+13}\mathrm{d}x}$.

By writing $2x+7=(2x+4)+3$, find ${\displaystyle \int h(x)\mathrm{d}x}$.

Hence find the exact value of ${\displaystyle {\int }_{-2}^{1}h(x)\mathrm{d}x}$, giving your answer in the form $\ln a+{\displaystyle \frac{\pi }{b}}$, where $a,b\in {\mathbb{Z}}^{+}$.

The region bounded by the curve $y={\displaystyle \frac{3}{\sqrt{x^2+4x+13}}}$, the $x$-axis and the lines $x=-2$ and $x=1$ is rotated through $2\pi$ radians about the $x$-axis.

Find the exact volume of the solid formed.

Show that $g$ is an even function.

By considering limits, show that the graph of $y=g(x)$ has a horizontal asymptote and state its equation.

Find the coordinates of the maximum point on the graph of $y=g(x)$.

Hence write down the range of $g$.

Solve the equation $g(x)=1$.

Find the area of the region enclosed by the graph of $y=g(x)$, the $x$-axis and the lines $x=-2$ and $x=2$.

The region described in part (f) is rotated $2\pi$ about the $x$-axis. Find the volume of the solid formed.

Find the coordinates of the points of inflexion on the graph of $y=g(x)$.

Show that $A_n={\displaystyle \frac{1}{n+1}}$.

The region $R_n$ is rotated by $2\pi$ radians about the $x$-axis to form a solid of volume $X_n$.

Show that $X_n={\displaystyle \frac{\pi }{2n+1}}$.

The region $R_n$ is rotated by $2\pi$ radians about the $y$-axis to form a solid of volume $Y_n$.

By considering a cylinder of radius $1$ and height $1$, show that $Y_n={\displaystyle \frac{2\pi }{n+2}}$.

Show that $Y_n>X_n$ for all $n\in {\mathbb{Z}}^{+}$.

For a region under a curve $y=f(x)\ge 0$, for $a\le x\le b$, with area $A$, the centroid is the point $(\bar{x},\bar{y})$ where

Show that the centroid of $R_n$ is $({\displaystyle \frac{n+1}{n+2}},{\displaystyle \frac{n+1}{2(2n+1)}})$.

Pappus's theorem states that when a region of area $A$ is rotated by $2\pi$ radians about a line that does not pass through its interior, the volume of the solid formed is $V=2\pi dA$, where $d$ is the distance from the centroid of the region to the line.

Verify Pappus's theorem for $R_n$ rotated about the $x$-axis.

Verify Pappus's theorem for $R_n$ rotated about the $y$-axis.

Consider the disc $D$ defined by $x^2+(y-R{)}^2\le r^2$, where $0<r<R$. The disc $D$ is rotated by $2\pi$ radians about the $x$-axis to form a solid ring (a torus).

Use Pappus's theorem to write down the volume of the torus in terms of $r$ and $R$.

By considering the curves $y=R+\sqrt{r^2-x^2}$ and $y=R-\sqrt{r^2-x^2}$, use integration to confirm your answer to part (g)(i).

The semicircular region bounded by $y=\sqrt{r^2-x^2}$ and the $x$-axis is rotated by $2\pi$ radians about the $x$-axis to form a sphere.

Use Pappus's theorem to find the distance of the centroid of this semicircular region from the $x$-axis.

Show that the $x$-coordinate of P is ${\displaystyle \frac{\pi }{3}}$.

Find the gradient of each curve at the origin, and hence state which curve lies above the other for $0<x<{\displaystyle \frac{\pi }{3}}$.

The region $R$ is enclosed by the two curves between the origin and P.

Show that the area of $R$ is $1-\ln 2$.

The region $R$ is rotated through $2\pi$ radians about the $x$-axis.

Find the exact volume of the solid formed.

Find the exact coordinates of the point where the two graphs intersect.

Find the area of $R$.

The region $R$ is rotated $2\pi$ about the $x$-axis. Find the volume of the solid formed.

The region $R$ is rotated $2\pi$ about the $y$-axis.

Show that the volume of the solid formed is ${\displaystyle \frac{\pi }{2}}$.

Use integration to show that the volume of the cone is $V={\displaystyle \frac{1}{3}}\pi r^{2}h$.

The interval $0\le x\le h$ is divided into $n$ equal parts. On the $k$th part, ${\displaystyle \frac{(k-1)h}{n}}\le x\le {\displaystyle \frac{kh}{n}}$, an outer cylinder is drawn with radius equal to the value of $y$ at $x={\displaystyle \frac{kh}{n}}$. The total volume of the $n$ outer cylinders is $U_n$.

Show that $U_n={\displaystyle \frac{\pi r^{2}h}{n^3}}{\displaystyle \sum _{k=1}^n{k}^2}$.

Use mathematical induction to prove that ${\displaystyle \sum _{k=1}^n{k}^2=\frac{n(n+1)(2n+1)}{6}}$, for $n\in {\mathbb{Z}}^{+}$.

Hence, show that $U_n={\displaystyle \frac{\pi r^{2}h(n+1)(2n+1)}{6n^2}}$.

On each part, an inner cylinder is also drawn, with radius equal to the value of $y$ at $x={\displaystyle \frac{(k-1)h}{n}}$. The total volume of the $n$ inner cylinders is $L_n$.

Show that $L_n={\displaystyle \frac{\pi r^{2}h(n-1)(2n-1)}{6n^2}}$.

Show that ${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }{U}_n=\underset{n\to \mathrm{\infty }}{\lim }{L}_n=V}$.

Show that $U_n-L_n={\displaystyle \frac{\pi r^{2}h}{n}}$.

Find the least value of $n$ for which $U_n-L_n$ is less than $1\mathrm{\%}$ of $V$.

A paraboloid is formed by rotating the curve $y=r\sqrt{{\displaystyle \frac{x}{h}}}$, for $0\le x\le h$, by $2\pi$ radians about the $x$-axis. Outer and inner cylinders are drawn in the same way as for the cone.

Show that, for the paraboloid, $U_n-L_n={\displaystyle \frac{\pi r^{2}h}{n}}$ also.

Suggest why $U_n-L_n={\displaystyle \frac{\pi r^{2}h}{n}}$ for any solid formed by rotating an increasing curve from $(0,0)$ to $(h,r)$ about the $x$-axis.