Applications of integration

(i) Sketch $y=2/[1+(x-2)^2]$ and $y=1$ on the same diagram, showing clearly the coordinates of the stationary point and intersection points.

Curve $C_1$: $y=1/x$; curve $C_2$: $y=1/\sqrt{2-x^2}$. Region Q is bounded by $x=1/\sqrt2$ and both curves.

Curve C: $y=3-10/(x^2-2x+4)$.

C: $y=(x^2+5x+3)/(x+1)$. S is bounded by C, the $y$-axis and $y=9/2$. Find the volume when S is rotated completely about the $x$-axis.

Find the volume when the area bounded by $y=(x-1)/(x+1)$, the $x$-axis and $x=2,x=3$ is rotated completely about the $y$-axis. Give your answer to 3 significant figures.

(i) R is bounded by the ellipse $x^2/a^2+y^2/b^2=1$ and the line $y/b-x/a=1$, as shown. Show that its area is

It is given that

C: $x=t-\sin t$, $y=\sin^2t$, $0\le t\le\pi$. A has parameter $\pi/2$ and is a maximum point. Line l passes through the origin and A. R is bounded by C and l as shown.

C: $x=3t^2$, $y=t^3+3t$, where $t$ is real. A line l cuts C at two points P and Q with parameters $t=1$ and $t=-1/2$, respectively, as shown. Find the area bounded by C and l to 3 significant figures.

C: $x=-1/t$, $y=1/\sqrt{4-t^2}$, $-2<t<0$. S is bounded by C, $x=1/\sqrt3,x=1/\sqrt2$ and the $x$-axis.

(a) $x=2t^2$, $y=\sin(-t)$, $t\ge0$. Find the exact area bounded by curve, $x$-axis and $x=\pi^2/8$.

(a) Find $\int(\cot^62x+\cot^42x-\sin(x/2)\sin(3x/2))dx$.

(a) C: $y=e^{-x}/(1+e^{-x})$. R is bounded by C, positive $x$-axis and positive $y$-axis as shown; graph labels $y=1,y=0,1/2,R,O$. (i) Obtain a formula in terms of $n$ for $\int_0^n\dfrac{e^{-x}}{1+e^{-x}}dx$, $n>0$. (ii) Hence find R’s exact area. (iii) Denoting the answer obtained in part (i) by $q$, write down in terms of $q$ the area of R if C is…

The graph $y=2^x$, $0\le x\le1$, is shown. Rectangles, each width $1/n$, where $n$ is an integer, are drawn under the curve. Show their total area is $1/[n(2^{1/n}-1)]$.

(i) Prove $\dfrac d{dx}(\tan^{-1}x)=1/(1+x^2)$.