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Chapter 6: Q. 14 (page 574)

Set up and solve definite integrals to find each volume, surface area, or arc length that follows. Solve each volume problem both with disks/washers and with shells, if possible.
The centroid of the region between the graph of $f (x) = x^{2}$ and the $x$ -axis on $[0, 2]$ .

Short Answer

Expert verified

$\overset{炉}{x} = \frac{3}{2}, \overset{炉}{y} = \frac{6}{5}$ .

Step by step solution

01

Step 1. Given Information.

The centroid of the region between the graph of $f (x) = x^{2}$ and the $x$ -axis on $[0, 2] .$

02

Step 2. Calculation of a mass and moments.

The total mass is :

$m = 蚁 {鈭�}_{a}^{b} f (x) d x m = {鈭�}_{0}^{2} x^{2} d x m = {[\frac{x^{3}}{3}]}_{0}^{2} m = \frac{8}{3}$

The moments are :

$M_{x} = 蚁 {鈭�}_{a}^{b} \frac{{[f (x)]}^{2}}{2} d x M_{x} = \frac{1}{2} {鈭�}_{0}^{2} x^{4} d x M_{x} = {[\frac{x^{5}}{2 脳 5}]}_{0}^{2} M_{x} = \frac{16}{5}$

$M_{y} = 蚁 {鈭�}_{a}^{b} x f (x) d x M_{y} = {鈭�}_{0}^{2} x^{3} d x M_{y} = {[\frac{x^{4}}{4}]}_{0}^{2} M_{y} = \frac{16}{4} = 4$

03

Step 3. Centroid of a region.

We know that

$\overset{炉}{x} = \frac{M_{y}}{m} = \frac{4}{\frac{8}{3}} = \frac{3}{2} \overset{炉}{y} = \frac{M_{x}}{m} = \frac{\frac{16}{5}}{\frac{8}{3}} = \frac{16 脳 3}{5 脳 8} = \frac{6}{5} .$

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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.

29. $\frac{d y}{d x} = 3 y, y (0) = 4$

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

$\frac{d y}{d x} = 9.8 - \frac{0.3}{150} y, y (0) = 1000$

Consider the region between $f (x) = \sqrt{x}$ and the x-axis on $[0, 4]$ . For each line of rotation given in Exercises 27鈥�30, use four disks or washers based on the given rectangles to approximate the volume of the resulting solid.

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

$\frac{d y}{d x} = 10 - 7 y, y (0) = \frac{1}{11}$

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

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

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