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Chapter 6: Q 61 (page 524)

Use definite integrals to find the volume of each solid of revolution described in Exercises $49 - 61$ . (It is your choice whether to use disks/washers or shells in these exercises.)
The region bounded by the graphs of $f (x) = |x|$ and $y = 2$ on $[- 2, 2]$ , revolved around the line $x = 3$ .

Short Answer

Expert verified

The required volume by using shells is $V = 24 蟺$

Step by step solution

01

Step 1. Given Information

We have given the following function :-

$f (x) = |x|$ .

We have to find the volume of region of graph of this function and $y = 2$

02

Step 2. Find the integral and evaluate it to calculate volume

We know that by using shells the volume is given by :-

$V = 2 蟺 {鈭�}_{d}^{d} r (x) h (x) dx$ .

Here axis of revolution is $x = 3$ . So that $r (x) = 3 - x$ .

Also from role="math" localid="1651676255913" $y = - 2 t o 0$ height is given by $h (x) = 2 + x$

and from $y = 0 t o 2$ height is given by $h (x) = 2 - x$ .

So the volume is given by following :-

$V = 2 蟺 {鈭�}_{- 2}^{0} (3 - x) (2 + x) dx + 2 蟺 {鈭�}_{0}^{2} (3 - x) (2 - x) dx 鈬� V = 2 蟺 {鈭�}_{- 2}^{0} (6 + x - x^{2}) dx + 2 蟺 {鈭�}_{0}^{2} (6 - 5 x + x^{2}) dx 鈬� V = 2 蟺 {[6 x + \frac{x^{2}}{2} - \frac{x^{3}}{3}]}_{- 2}^{0} + 2 蟺 {[6 x - \frac{5 x^{2}}{2} + \frac{x^{3}}{3}]}_{0}^{2} 鈬� V = 2 蟺 [(0 + 0 - 0) - (- 12 + 2 + \frac{8}{3})] + 2 蟺 [(12 - 10 + \frac{8}{3})] 鈬� V = 2 蟺 [12 - 2 - \frac{8}{3} + 12 - 10 + \frac{8}{3}] 鈬� V = 2 蟺 [12] 鈬� V = 24 蟺$

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Most popular questions from this chapter

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29鈥�52

$\frac{d y}{d x} = e^{x + y}, y (0) = 2$

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.

$\frac{d y}{d x} = \frac{x + 1}{\sqrt{x}}$

Use antidifferentiation and/or separation of variables to solve the given differential equations. Your answers will involve unsolved constants.

$\frac{d y}{d x} = \frac{3 x + 1}{x y}$

Use antidifferentiation and/or separation of variables to solve each of the initial-value problems in Exercises 29-52.

$\frac{d y}{d x} = x y + x + y + 1, y (0) = c$

For each pair of definite integrals in exercise 13-18 decide which if either is larger without computing the integrals

$蟺 {鈭�}_{0}^{2} x^{4} d x a n d 蟺 {鈭�}_{0}^{2} (x^{4} - 16) d x$

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