Chapter 8.2, Problem 4QR

To determine

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

Answer to Problem 4QR

The area between $x-axis$ and $y=4x-x^3$ over $\left[0,2\right]$ $=4$

Explanation of Solution

Given information:

Function

  $y=4x-x^3$ over $\left[0,2\right]$

Concept used: If $f(x)$ and $g(x)$ are continuous with $f(x)\ge g(x)$ throughout $[a,b]$ , then the area between the curves $y=f(x)$ and $y=g(x)$ from a to b is the integral of $[f(x)-g(x)]$ from a to b

  $A={\displaystyle \underset{a}{\overset{b}{\int }}[f(x)-g(x)]}dx$

Calculation:

Function

  $y=4x-x^3$ over $\left[0,2\right]$

  $\begin{array}{l}A={\displaystyle \underset{0}{\overset{2}{\int }}\left[4x-x^3\right]}dx\\ A={\left[2x^2-\frac{x^4}{4}\right]}_0^2\\ A=4\end{array}$

The area between $x-axis$ and $y=4x-x^3$ over $\left[0,2\right]$ $=4$