Chapter 1.7, Problem 127E

(a)

To determine

To calculate:The value of area for $a=0\text{ and }b=1$. The area of region bounded by the graphs of $y=0$, $y=\frac{1}{\left(x^2+1\right)}$, $x=a$, and $x=b$ is shown in the below figure.

The formula for the area as $\text{Area}=\arctan b-\arctan a$.

(b)

To determine

To calculate: The value of area for $a=-1\text{ and }b=1$. The area of region bounded by the graphs of $y=0$, $y=\frac{1}{\left(x^2+1\right)}$, $x=a$, and $x=b$ is shown in below figure.

The formula for the area as, $\text{Area}=\arctan b-\arctan a$.

(c)

To determine

To calculate: The value of area for $a=0\text{ and }b=3$. The area of region bounded by the graphs of $y=0$, $y=\frac{1}{\left(x^2+1\right)}$, $x=a$, and $x=b$ shown in below figure.

The formula for the area as, $\text{Area}=\arctan b-\arctan a$.

(d)

To determine

To calculate: The value of area for $a=0\text{ and }b=3$. The area of region bounded by the graphs of $y=0$, $y=\frac{1}{\left(x^2+1\right)}$, $x=a$, and $x=b$ is shown in the below figure.

The formula for the area as, $\text{Area}=\arctan b-\arctan a$.