Practice section 4 of 9

The region $R$ is bounded by $C:y=e^{x^2}$ and $y=4$.

(i) Find the exact volume when $R$ is rotated through two right angles about the y-axis.

(ii) Write down the equation of the curve obtained when $C$ is translated four units in the negative y-direction. Hence find the approximate volume of the solid formed when $R$ is rotated completely about $y=4$.

Find the value of the volume of the solid generated when the region enclosed between $y=x^2$ and $y=\sqrt x$ is rotated about the x-axis through $2\pi$ radians.

Find the value of the volume of the solid formed by rotating the region between $y=x^2$ and $y=x^3$ about the y-axis.

The graph shows $y=1/(1+x^2)$ and $y=\tan x$, where the source prints $-\pi/2\le x\le\pi/2$.

(i) Find the area bounded by the two curves and $x=1$.

(ii) Find the volume generated when the region bounded by the two curves and the y-axis is rotated by $2\pi$ radians about the y-axis.

(i) Using $x=\sin\theta$, show that $\int_0^{1/2}\sqrt{1-x^2}\,dx=\pi/12+\sqrt3/8$. (ii) The circle $x^2+(y-1)^2=1$ and the line $y=1-\sqrt3x$ intersect at $A=(1/2,1-\sqrt3/2)$. Using part (i), find the exact area of the shaded region bounded by the circle, line and x-axis.

Find $k$ such that $\int_{-c}^0|x-c|\,dx=k\int_0^{2c}|x-c|\,dx$, where $c$ is a positive constant.