Chapter 8, Problem 2P

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336

Chapter
Section

### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270336
Textbook Problem

# Find the centroid of the region enclosed by the loop of the curve y2 = x3 − x4.

To determine

The centroid of the region enclosed by the curve y2=x3x4 .

Explanation

Given information:

The curve function is y2=x3âˆ’x4 (1)

Calculation:

Rearrange Equation (1) as shown below:

y=Â±x3âˆ’x4

Therefore, the loop of the curve have the functions of y=x3âˆ’x4 and y=âˆ’x3âˆ’x4 .

Draw the loop of the curve for the function y2=x3âˆ’x4 using the procedure as shown below:

• Draw the graph for the function y=x3âˆ’x4 by substituting different values for x.
• Similarly in the same graph plot for the function y=âˆ’x3âˆ’x4 by substituting different values for x.

The region enclosed by the curves y=x3âˆ’x4 and y=âˆ’x3âˆ’x4 is shown in Figure 1.

Refer Figure 1.

The loop of the curve is symmetric about x-axis. Therefore the y-coordinate of the centroid is yÂ¯=0 .

The total area of the curve is twice the area of the top curve.

The interval of the loop of the curve is x=0 to x=1 .

The expression to find the area of the shaded region is shown below:

A=âˆ«ab[f(x)]â€‰dx (2)

Here, the lower limit is a, the upper limit is b, and the top curve function is f(x) , and the bottom curve function is g(x) .

Rearrange Equation (2) for the curve being symmetric.

A=2âˆ«ab[f(x)]â€‰dx (3)

Substitute 0 for a, 1 for b, and x3âˆ’x4 for f(x) in Equation (3).

A=2âˆ«01(x3âˆ’x4)â€‰dx=2âˆ«01(x321âˆ’x)â€‰dx (4)

Let sinÎ¸=x (5)

Find the limit of Î¸ :

Substitute 0 for x in Equation (5).

sinÎ¸=0Î¸=0

Substitute 1 for x in Equation (5).

sinÎ¸=1sinÎ¸=1Î¸=Ï€2

Substitute sinÎ¸ for x and change the limits as 0â€‰toâ€‰Ï€2 in Equation (4).

A=2âˆ«0Ï€2(2sin4Î¸cosÎ¸1âˆ’sin2Î¸)â€‰dÎ¸=2âˆ«0Ï€2(2sin4Î¸cosÎ¸cos2Î¸)â€‰dÎ¸=2âˆ«0Ï€2(2sin4Î¸cos2Î¸)â€‰dÎ¸=4âˆ«0Ï€2(14(1âˆ’cos2Î¸)212(1+cos2Î¸))â€‰dÎ¸

=12âˆ«0Ï€2(1âˆ’cos2Î¸âˆ’cos22Î¸+cos32Î¸)â€‰dÎ¸=12âˆ«0Ï€2(1âˆ’cos2Î¸âˆ’12(1+cos4Î¸)+cos2Î¸(1âˆ’sin22Î¸))â€‰dÎ¸=12âˆ«0Ï€2(1âˆ’cos2Î¸âˆ’12âˆ’12cos4Î¸+cos2Î¸(1âˆ’sin22Î¸))â€‰dÎ¸=12âˆ«0Ï€2(12âˆ’12cos4Î¸âˆ’cos2Î¸sin22Î¸)â€‰dÎ¸ (6)

Integrate Equation (6) and apply the limits.

A=12[Î¸2âˆ’18sin4Î¸âˆ’16sin32Î¸]0Ï€2=12[(Ï€4âˆ’18sin4(Ï€2)âˆ’16sin32(Ï€2))âˆ’(0)]=12(Ï€4âˆ’0)=Ï€8

Calculate the x-coordinate of the centroid (xÂ¯) using the relation:

xÂ¯=1Aâˆ«abx[f(x)]â€‰dx (7)

Rearrange Equation (7) for the curve being symmetric

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