Chapter 4.7, Problem 125PE

In calculus, we can show that the area below the graph of $f\left(x\right)=\frac{1}{1+x^2}$ , above the $x\text{-}$ axis, and between the lines $x=a$ and $x=b$ for $a<b$ , is given by ${\tan }^{-1}b-{\tan }^{-1}a$

a. Find the area under the curve between $x=0$ and $x=1$ .

b. Evaluate $f\left(0\right)$ and $f\left(1\right)$ .

c. Find the area of the trapezoid defined by the points $\left(0,0\right),\left(1,0\right),\left[0,f\left(0\right)\right],$ and $\left[1,f\left(1\right)\right]$ to confirm that your answer from part (a) is reasonable.