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Chapter 4: Q. 37 (page 386)

For each function $f$ f and interval $[a, b]$ in Exercises 26–37, use definite integrals and the Fundamental Theorem of Calculus to find the exact values of
(a) the signed area and
(b) the absolute area of the region between the graph of f and the x-axis from localid="1649417613753" $x = a a n d x = b .$
$f (x) = \frac{1}{1 + x^{2}}, [a, b] = [- 1, 1]$

Short Answer

Expert verified

The answer is

Part (a) The signed area is $\frac{π}{2}$ .

Part (b) The absolute area is $\frac{π}{2}$ .

Step by step solution

01

Part (a) Step 1. Given Information.

The given function and interval is $f (x) = \frac{1}{1 + x^{2}}, [a, b] = [- 1, 1]$ .

02

Part (a) Step 2. Explanation .

The signed area in the interval will be,

$\int_{- 1}^{1} (\frac{1}{1 + x^{2}}) d x = \tan^{- 1} (x) = \tan^{- 1} (1) - \tan^{- 1} (- 1) = \frac{π}{4} + \frac{π}{4} = \frac{2 π}{4} = \frac{π}{2}$

03

Part (b) Step 1. Graph of the function

The graph of the function is,

04

Part (b) Step 2. Absolute area.

The absolute area will be,

$\int_{- 1}^{1} (\frac{1}{1 + x^{2}}) d x = \tan^{- 1} (x) = \frac{π}{4} + \frac{π}{4} = \frac{π}{2}$

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Most popular questions from this chapter

Show by exhibiting a counterexample that, in general, $\int f (x) g (x) d x \neq (\int f (x) d x) (\int g (x) d x)$ . In other words, find two functions f and g so that the integral of their product is not equal to the product of their integrals.

Describe an example that illustrates that $\int_{a}^{b} | f (x) | d x$ is not equal to $| \int_{a}^{b} f (x) d x |$ .

Compare the definitions of the definite and indefinite integrals. List at least three things that are different about these mathematical objects.

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess- and- check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating. (Hint for Exercise 54: $\tan (x) = \frac{\sin (x)}{\cos (x)}$ ).

$\int (\sec^{2} (x) + \csc^{2} (x)) d x .$

Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.

$\int \frac{3}{x^{2} + 1} d x$

See all solutions

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