Integrating the exponential over the area bounded by the functions $y=x$ and $y=x^3$

Question

Can someone please help me solve the following problem below? Thank you

Compute the integral of the function over the area bounded by the functions $y=x$ and $y=x^3$ $$f(x,y) = e^{x^2}$$

Answer 1

HINT

You can think about it in terms of physics. If the function $f(x,y)$ denotes the mass distribution, the mass corresponding to the area between the curves $y = x$ and $y = x^{3}$ is given by \begin{align*} M = \int_{D}\mathrm{d}m = \int_{D}f(x,y)\mathrm{d}x\mathrm{d}y \end{align*}

At your case, $D = \{(x,y)\in\textbf{R}^{2}\mid (0\leq x\leq 1)\wedge(x^{3}\leq y\leq x)\}$.

Hence we have the following result: \begin{align*} \int_{0}^{1}\int_{x^{3}}^{x}e^{x^{2}}\mathrm{d}y\mathrm{d}x = \int_{0}^{1}(x - x^{3})e^{x^{2}}\mathrm{d}x \end{align*}

Can you take it from here?