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}`