Find $\displaystyle\int x^2 e^{-2x} \, \mathrm{d}x$.[4] The curve with equation $y=xe^{-x}$ and

Question

Find $\displaystyle\int x^2 e^{-2x} \, \mathrm{d}x$.[4] The curve with equation $y=xe^{-x}$ and the line with equation $y=\frac{1}{e}x$ meet at the origin $O$ and the point $P$ with $x$-coordinate $1$. The region $R$ is bounded by the curve and the line (see diagram). Find the exact volume of the solid formed when $R$ is rotated through $360{}^\circ$ about the $x$-axis.[4]

Accepted Answer

(a)$-\frac{1}{4}\mathrm{e}^{-2x} \left[2x^2 + 2x + 1\right] + C$ (b)$\frac{\pi}{12}(3-19e^{-2})$units$^3$

2025 JPJC P1 Q6

2025 JPJC P1 Q6