Question

Areas and average values: Set up and solve definite integrals to calculate the exact values of each of the given geometric quantities. Verify that your answers are reasonable with graphs.

The signed area between the graph of $f\left(x\right)=\frac{x^2-1}{x^2+1}$and the x-axis on the interval$[-4,4]$.

Short answer

The area between the curve is $2.69$.

Step-by-step solution

Given information

The given function is$f\left(x\right)=\frac{x^2-1}{x^2+1}$

Simplify the given function

Rewrite the given function.

Divide $x^2-1$by $x^2+1$using the long division method.

The quotient is $1$and the remainder is $-2$.

$f\left(x\right)=1-\frac{2}{x^2+1}$

Determine the area of the given region

$\begin{array}{rcl}{\int }_{-4}^{4}f\left(x\right)dx& =& {\int }_{-4}^4\left(1-\frac{2}{x^2+1}\right)dx\\ & =& x{|}_{-4}^4-2\left({\tan }^{-1}x{|}_{-4}^4\right)\\ & =& (4-(-4\left)\right)-2\left({\tan }^{-1}\right(4)-{\tan }^{-1}(-4\left)\right)\\ & =& 8-4{\tan }^{-1}\left(4\right)\\ & =& 2.69\end{array}$

Draw the graph of the given function in the given interval.

The graph is shown below:

Conclusion

The area is$2.69$.