Integration

Use the trapezium rule with three intervals to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures.

The shaded region is rotated completely about the $x$-axis.

Find the exact volume of the solid produced.

Use the trapezium rule with two intervals to find an approximation to the area of the shaded region. Give your answer correct to 2 significant figures.

The shaded region is rotated completely about the $x$-axis.

Find the exact volume of the solid produced.

Use the trapezium rule with two intervals to find an approximation to the area of region $A$. Give your answer correct to 3 significant figures.

Find the exact total area of regions $A$ and $B$. Give your answer in the form $k \ln m$, where $k$ and $m$ are constants.

Deduce an approximation to the area of region $B$. Give your answer correct to 3 significant figures.

State, with a reason, whether your answer to part (c) is an over-estimate or an under-estimate of the area of region $B$.