Chapter 6.4, Problem 70E

To determine

To calculate: The area of the region enclosed between $f\left(x\right)=\sqrt{1-x^4}$ and the x-axis

Answer to Problem 70E

The area under curve is 1.748

Explanation of Solution

Given information:

The area of the region enclosed between the graph of $f\left(x\right)=\sqrt{1-x^4}$ and the x-axis

Formula used:

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx=\text{NINT}\left(f\left(x\right),x,a,b\right)$

Calculation:

Correct option is (E).

Calculator has a numerical integration capability, which will denote as NINT. In particular, using $\text{NINT}\left(f\left(x\right),x,a,b\right)$ to denote a calculator approximation of ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx$ . Hence,

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx=\text{NINT}\left(f\left(x\right),x,a,b\right)$

The understanding that the right-hand side of the equation is an approximation of the left-hand side.

Now, use above expression with this function

  $f\left(x\right)=\sqrt{1-x^4}$

Since, it must be

  $\begin{array}{l}1-x^4\ge 0\\ x^4\le 1\\ -1\le x\le 1\end{array}$

So,

  $\begin{array}{l}\left[a,b\right]=\left(-1,1\right)\\ \text{area under curve}={\displaystyle \underset{-1}{\overset{1}{\int }}\left(\sqrt{1-x^4}\right)dx}\\ =\text{NINT}\left(\sqrt{1-x^4},x,-1,1\right)\\ \text{area under curve}=1.748\end{array}$

Conclusion:

The area under curve is 1.748