Question

Find the area between the graph of f of x equals the product of x squared and e raised to negative 1 times x cubed power and the x-axis for the interval (0, ∞). Your work must include the proper notation and show the antiderivative. If the integral diverges, show why.

Answer 1

If $f(x)=x^2e^{-x^3}$, then the area between the graph of $f(x)$ and the x-axis for non-negative x is given by the integral,

$\displaystyle\int_0^\infty x^2e^{-x^3}\,\mathrm dx$

Let $u=-x^3$ and $\mathrm du=-3x^2\,\mathrm dx$; then the integral is equivalent to

$\displaystyle-\frac13\int_0^{-\infty}e^u\,\mathrm du=\frac13\int_{-\infty}^0e^u\,\mathrm du=\frac13\left(1-\lim_{u\to-\infty}e^u\right)=\boxed{\frac13}$