Answer
(a)
(i) the area of the surface obtained by rotating the curve about the x-axis
$$
S=\int 2 \pi y d s=\int_{-1}^{1} 2 \pi e^{-x^{2}} \sqrt{1+4 x^{2} e^{-2 x^{2}}} d x
$$
(ii) the area of the surface obtained by rotating the curve about the y-axis
$$
S=\int 2 \pi x d s=\int_{-1}^{1} 2 \pi x \sqrt{1+4 x^{2} e^{-2 x^{2}}} d x
$$
(b) the surface areas, correct to four decimal places, by using the numerical integration, are:
(i) 11.0753
(ii) 3.9603
Work Step by Step
$$
y= e^{-x^{2}}, \quad \quad -1 \leq x \leq 1
$$
$\Rightarrow$
$$
y^{\prime} =e^{-x^{2}}. (-2x)
$$
$\Rightarrow$
$$
\begin{aligned} ds &= \sqrt {1+\left(y^{\prime}\right)^{2}} dx \\
&=\sqrt{1+4x^{2}e^{-2x^{2}}} dx
\end{aligned}
$$
(a)
(i) the area of the surface obtained by rotating the curve about the x-axis
$$
S=\int 2 \pi y d s=\int_{-1}^{1} 2 \pi e^{-x^{2}} \sqrt{1+4 x^{2} e^{-2 x^{2}}} d x
$$
(ii) the area of the surface obtained by rotating the curve about the y-axis
$$
S=\int 2 \pi x d s=\int_{-1}^{1} 2 \pi x \sqrt{1+4 x^{2} e^{-2 x^{2}}} d x
$$
(b) the surface areas, correct to four decimal places, by using the numerical integration, are
(i) 11.0753
(ii)3.9603
