The diagram shows the region $T$, which is bounded by the curve $y = e^{x}$, the line $y = 2$,

Question

The diagram shows the region $T$, which is bounded by the curve $y = e^{x}$, the line $y = 2$, and the $y$-axis and the region $U$, which is bounded by $y = e^{x}$, the axes and the line $y = \ln 2$. Find the volume of solid formed when $T$ is rotated through $4$ right angles about the $y$-axis. Find the volume of solid formed exactly when $U$ is rotated through $2\pi$ about the $x$-axis. Find the volume of solid formed exactly when $T$ is rotated through $4$ right angles about $x$-axis. Find the volume of solid formed when $U$ is rotated through $4$ right angles about the $y$-axis.

Accepted Answer

(i) $\pi\left(2(\ln2)^2-4\ln2+2\right)=0.592\text{ units}^3\text{ (3 s.f.)}$ (ii) $\frac{3\pi}{2}\text{ units}^3$ (iii) $\pi\left(4\ln2-\frac32\right)\text{ units}^3$ (iv) $\pi(4\ln2-2)=2.43\text{ units}^3\text{ (3 s.f.)}$

Applications of Integration - Volume of Revolution

Applications of Integration - Volume of Revolution

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