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Chapter 6: Q. 37 (page 556)

In Exercises 35鈥�40, use definite integrals to calculate the centroid of the region described. Use graphs to verify that your answers are reasonable.
The region between f(x)=x3 and the line y = 8 on [a, b] = [0, 2]. (Compare with Exercise 33.)

Short Answer

Expert verified

The centroid of the region between f(x)=x3and y=8 is (0,0.2).

Step by step solution

01

Step 1. Given Information.

The function:

f(x)=x3y=8on[0,2]

02

Step 2. Centroid of region under curves.

The centroid of the region between the curve f(x) and x-axis is:

(x炉,y炉)=(鈭�abxf(x)dx鈭�abf(x)dx,鈭�abf(x)2dx鈭�abf(x)dx)

03

Step 3. Find the denominator.

鈭�abf(x)dx=鈭�ab(x3-8)dx=[x44-8x]02=12鈭�abxf(x)dx=鈭�abx(x3-8)dx=[x55-4x2]02=9.6

04

Step 4. Find ∫abf(x)2dx

鈭�abf(x)2dx=鈭�ab(x6)-8)dx=2.29

05

Step 5. Substitute the known values in the formula.

(x炉,y炉)=(鈭�abxf(x)dx鈭�abf(x)dx,鈭�abf(x)2dx鈭�abf(x)dx)=(9.612,2.2912)=(0.08,0.19)=(0,0.2)

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Most popular questions from this chapter

Sketching a representative disks ,washers and shells : sketch a representative disks , washers , shells for the solid obtained by revolving the regions shown in figure around the given lines .

the linex=7

The volume of solid obtained by revolving the region between the graph f(x)=xandg(x)=x3on[0,1]around (a)the y axis (b)the line x=2

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