Chapter 7.2, Problem 4QR

To determine

To find: The area between the $x$ -axis and the graph of the given function over the given interval.

Answer to Problem 4QR

The answer is: $A=4$

Explanation of Solution

Given information:

The function is $y=4x-x^3$ over $[0,2]$ .

Calculation:

  $\begin{array}{c}A={\int }^{\text{}}{}_0^2\left(4x-x^3-0\right)\text{d}x=4{\int }^{\text{}}{}_0^{2}x\text{d}x-{\int }^{\text{}}{}_0^2{x}^3\text{d}x\\ ={\left.4\cdot \frac{x^2}{2}\right|}_0^2-{\left.\frac{x^4}{4}\right|}_0^2\\ =2\left(2^2-0\right)-\frac{1}{4}\left(2^4-0\right)\\ =8-4\\ =4\end{array}$

  

Therefore, the required area between the $x$ -axis and the graph of the given function over the given interval is $A=4$ .