Answer
a.) $V=3.75825$
b.) $V=13.14312$
Work Step by Step
a.)
$\displaystyle{A(x)=\pi\left(e^{-x^2}\right)^2}\\ \displaystyle{A(x)=\pi\left(e^{-2x^2}\right)}$
$\begin{aligned} V &=\int_{-1}^{1} A(x) \ d x \\ V &=\int_{-1}^{1} \pi\left(e^{-2x^2}\right) \ d x \\ V &=\pi \int_{-1}^{1} e^{-2x^2}\ dx \\
V&=3.75825 \end{aligned}$
b.)
$\displaystyle{A(x)=\pi\left(1+e^{-x^2}\right)^2 - \pi\left(1\right)^2} \\ \displaystyle{A(x)=\pi\left(e^{-2x^2}+2e^{-x^2}+1\right) - \pi}\\
\displaystyle{A(x)=\pi\left(e^{-2x^2}+2e^{-x^2}\right)}$
$\begin{aligned} V &=\int_{-1}^{1} A(x) \ d x \\ V &=\int_{-1}^{1} \pi\left(e^{-2x^2}+2e^{-x^2}\right) \ d x \\ V &=\pi \int_{-1}^{1} e^{-2x^2}+2e^{-x^2}\ dx \\
V&=13.14312\end{aligned}$
