Using the substitution ${{u}^{2}}=x+1$, evaluate \(\displaystyle\displaystyle \displaystyle\int_{-

Question

Using the substitution ${{u}^{2}}=x+1$, evaluate $\displaystyle\displaystyle \displaystyle\int_{-1}^{a-1}{x\sqrt{x+1}\ \mathrm{d}x}$, giving your answer in terms of $a$.[5] The curve $C$ has equation $y=x\sqrt{x+1}$. The region $R$ is bounded by $C$, the lines $y=\sqrt{2}$ and $x=-1$. Find the exact volume of the solid obtained when $R$ is rotated through $2\pi $ radians about the line $y=\sqrt{2}$.[7]

Accepted Answer

(i) $\dfrac25a^{5/2}-\dfrac23a^{3/2}$ (ii) $\dfrac{18\pi}{5}\text{ units}^3$

2023 RVHS P2 Q5

2023 RVHS P2 Q5

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