Chapter 5.2, Problem 59E

To determine

To find: The integral ${{\displaystyle \int }}_{-1}^1{x}^{3}dx$

Answer to Problem 59E

The integral is $0$ .

Explanation of Solution

Given information:

  ${{\displaystyle \int }}_0^1{x}^{3}dx=\frac{1}{4}$

And the integration is ${{\displaystyle \int }}_{-1}^1{x}^{3}dx$

Calculation:

Required to calculate the value of the provided integral in this problem using the value of:

  ${{\displaystyle \int }}_0^1{x}^{3}dx=\frac{1}{4}$

And by looking at the integrand's graph. Therefore,  approach will be to graph the integrand and then attempt to calculate the integral's value by examining the graph.

Deal with the following integral:

  ${{\displaystyle \int }}_{-1}^1{x}^{3}dx$

The following is a graph of the integrand:

  

The positive and the negative area cancel out when  integrate since the graph is symmetrical around the origin point. The resulting integral is therefore 0.

Therefore the required integral of given integration ${{\displaystyle \int }}_{-1}^1{x}^{3}dx$ is $0$ .