Chapter 5.10, Problem 12E

To determine

To find whether the integral ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}(y^3-3y^2)dy}$ is convergent or divergent.

Answer to Problem 12E

The integral ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}(y^3-3y^2)dy}$ is divergent.

Explanation of Solution

Given information:

The integral is ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}(y^3-3y^2)dy}$

Concept used: If the value of the integral is finite, it is convergent otherwise it is divergent

Given, ${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}(y^3-3y^2)dy}$

On Integration, the integral becomes

  $\text{ =}{\left[\frac{y^4}{4}-3\frac{y^3}{3}\right]}_{-\infty }^{\infty }=\infty ,\text{infinite}\text{number}$

Since the given integral has infinite value.

Hence, the given integral is divergent