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Chapter 4: Q. 28 (page 399)

For each area accumulation function A in Exercises 27鈥�30,
(a) illustrate A(2) graphically,
(b) calculate A(2) and A(5), and
(c) find an explicit elementary formula for A(x).
$A (x) = {鈭�}_{- 蟺}^{x} \sin t . d t$

Short Answer

Expert verified

(a) The function A(2) is:

(b) The value of $A (2) = - (\cos 2 + 1) a n d A 95) = - (\cos 5 + 1)$ .

(c) The explicit elementary formula is $A (x) = - (\cos x + 1)$

Step by step solution

01

Part (a) Step 1. Given Information.

The function:

$A (x) = {鈭�}_{- 蟺}^{x} \sin t . d t$

02

Part (a) Step 2. Graph the function.

Graph the function, $A (x) = {鈭�}_{- 蟺}^{x} \sin t . d t$ .

So graph the function $A (x) = {鈭�}_{- 蟺}^{x} \sin t . d t$ between the points $x = - 蟺 to x = 2$ .

03

Part (b) Step 1. Calculate A(2).

$A (x) = {鈭�}_{- 蟺}^{x} \sin t . d t A (2) = {鈭�}_{- 蟺}^{2} \sin t . d t = {[- \cos t]}_{- 蟺}^{2} = [- \cos 2] - [- \cos (- 蟺)] = - (\cos 2 + 1)$

04

Part (b) Step 2. Calculate A(5).

$A (x) = {鈭�}_{- 蟺}^{x} \sin t . d t A (5) = {鈭�}_{- 蟺}^{5} \sin t . d t = {[- \cos t]}_{蟺}^{5} = [- \cos 5] - [- \cos 蟺] = - (\cos 5 + 1)$

05

Part (c) Step 1. Find an explicit formula.

Find an explicit formula for the given function.

$A (x) = {鈭�}_{- 蟺}^{x} \sin t . d t = {[- \cos t]}_{- 蟺}^{x} = - \cos x - (\cos (- 蟺)) = - \cos x - \cos 蟺 = - (\cos x + 1)$

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Most popular questions from this chapter

Use integration formulas to solve each integral in Exercises 21鈥�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: $\tan (x) = \frac{\sin (x)}{\cos (x)}$ ).

$鈭� \frac{2 x \ln x - x}{{(\ln x)}^{2}} d x .$

Write each expression in Exercises 41鈥�43 in one sigma notation (with some extra terms added to or subtracted from the sum, as necessary).

$3 {鈭�}_{k = 2}^{25} k^{2} + 2 {鈭� k}_{k = 2}^{24} - {鈭�}_{k = 0}^{25} 1$

Find the sum or quantity without completely expanding or calculating any sums.

Given ${鈭�}_{k = 3}^{10} a_{k} = 12$ and ${鈭�}_{k = 2}^{10} a_{k} = 23$ , find $a_{2}$ .

If ${鈭�}_{- 2}^{3} f (x) d x = 4, {鈭�}_{- 2}^{6} f (x) d x = 9, {鈭�}_{- 2}^{3} g (x) d x = 2$ , and ${鈭�}_{3}^{6} g (x) d x = 3$ , then find the values of each definite integral in Exercises $29 - 40$ . If there is not enough information, explain why.

${鈭�}_{- 2}^{- 2} x (f (x) + 3)^{2} d x$

Repeat Exercise 13 for the function f shown above at the right, on the interval $[- 2, 2]$

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