Chapter 15.4, Problem 8CFU

To determine

To find: the area between the curve and the x-axis for the given interval.

Answer to Problem 8CFU

  $\frac{32}{5}uni{t}^2$ .

Explanation of Solution

Given:

Thegiven intervalis $y=x^4$ from $x=0\text{ to }x=2$ .

Calculation:

The area is the integral from $0\text{ to }4$ of $x^4$

Thus, thearea is ${\displaystyle \underset{0}{\overset{2}{\int }}\left(x^4\right)dx}$ .

Now, use Fundamental Theorem of Calculus of evaluate. Then,

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{2}{\int }}\left(x^4\right)dx}\\ ={\left[\frac{x^{4+1}}{4+1}\right]}_0^2\left\{\because x^{n}dx=\frac{x^{n+1}}{n+1}\right\}\\ ={\left[\frac{x^5}{5}\right]}_0^2\\ =\left[\frac{{\left(2\right)}^5}{5}\right]-\left[\frac{{\left(0\right)}^5}{5}\right]\\ =\left[\frac{32}{5}\right]-\left[0\right]=\frac{32}{5}\end{array}$

Hence,the area between the curve and the x-axis for the given intervalis $\frac{32}{5}uni{t}^2$ .