Chapter 6.2, Problem 54E

To determine

To evaluate: the integral.

Answer to Problem 54E

The integral of the equation is $0$ .

Explanation of Solution

Giveninformation :

The fact:

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

The integral:

  ${{\displaystyle \int }}_{-8}^8{x}^{3}dx$

Calculation :

For any integral function,

  ${\displaystyle {\int }_a^{b}f}(x)dx=($ area above the $x$ -axis) $-($ area below the $x$ axis)

  

Since, observe the graph of $x^3,$ it is symmetrical about line $y=x$ .so, area above the x-axis

and area below the $\text{x}$ -axis are equal.

So,

  ${{\displaystyle \int }}_{-8}^8{x}^{3}dx$ =(area above the x-axis)−(area below the x-axis)

=0