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

Question

Find $\displaystyle \displaystyle\int x^2 e^{-2x} \, \mathrm{d}x$.[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) $-\frac14e^{-2x}(2x^2+2x+1)+C$. (b) $\frac\pi{12}(3-19e^{-2})$ cubic units.

2025 JPJC P1 Q6

2025 JPJC P1 Q6

Solution:

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