Question

using calculus, the area of the region under y = \frac{1}{1 + x^{2}}, above the x - axis and between x = a and x = b is \tan^{-1}b-\tan^{-1}a. use this result, shown in the figure, to find the area of the region under y = \frac{1}{1 + x^{2}}, above the x - axis, and between a = 1 and b = 2. the area is approximately square units (round to four decimal places as needed.)

Explanation:

Step1: Identify the formula

The formula for the area is $\tan^{- 1}b-\tan^{- 1}a$.

Step2: Substitute values

Substitute $a = 1$ and $b = 2$ into the formula: $\tan^{- 1}(2)-\tan^{- 1}(1)$.

Step3: Calculate inverse - tangent values

We know that $\tan^{- 1}(1)=\frac{\pi}{4}\approx0.7854$ and $\tan^{- 1}(2)\approx1.1071$.

Step4: Compute the area

$A=\tan^{- 1}(2)-\tan^{- 1}(1)\approx1.1071 - 0.7854=0.3217$.

Answer:

$0.3217$