Chapter 8.3, Problem 42E

### 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

# A rectangle  with sides a and b is divided into two parts  and  by an arc of a parabola that has its vertex at one corner of  and passes through the opposite corner. Find the centroids of both  and .

To determine

To find: The centroid of both part in the rectangle R1 and R2.

Explanation

Given:

The length of the rectangle is a.

The width of the rectangle is b.

Calculation:

Draw the rectangular as shown in Figure 1.

Refer to Figure 1.

The part in the rectangle (R1) is parabolaâ€‘1.

The part in the rectangle (R2) is parabolaâ€‘2.

Consider the area (A) of the rectangle (R) is,

A=ab

The equation of parabola is as shown,

y=kx2 (1)

Substitute a for x and b for y in Equation (1).

b=ka2k=ba2

Substitute ba2 for k in Equation (1).

y=ba2x2

Calculate the area (A1) of the parabolaâ€‘1:

A1=âˆ«abydx (2)

Substitute 0 for a, a for b, and ba2x2 for y in Equation (2).

A1=âˆ«0aba2x2dx=ba2âˆ«0ax2dx (3)

Integrate Equation (3).

A1=ba2[x2+12+1]0a=ba2[x33]0a=ba2(a33âˆ’0)=ab3

Calculate the (xÂ¯1) coordinate of centroid of the parabolaâ€‘1:

xÂ¯1=1A1âˆ«abxyâ€‰dx (4)

Substitute 0 for a, a for b, ab3 for A1, and ba2x2 for y in Equation (4).

xÂ¯1=1ab3âˆ«0ax(ba2x2)â€‰dx=3abÃ—ba2âˆ«0ax3â€‰dx=3a3âˆ«0ax3â€‰dx (5)

Integrate Equation (5).

xÂ¯1=3a3[x3+13+1]0a=3a3[x44]0a=3a3(a44âˆ’0)=34a

Calculate the (yÂ¯1) coordinate of centroid of the parabolaâ€‘1:

yÂ¯1=1A1âˆ«ab12[f(x)]2â€‰dx (6)

Substitute 0 for a, a for b, ab3 for A1, and ba2x2 for y in Equation (6).

yÂ¯1=1ab3âˆ«0a12(ba2x2)2â€‰dx=3abÃ—12Ã—b2a4âˆ«0ax4â€‰dx=3b2a5âˆ«0ax4â€‰dx (7)

Integrate Equation (7).

yÂ¯1=3b2a5[x4+14+1]0a=3b2a5[x55]0a=3b2a5(a55âˆ’0)=310b

Hence, the centroid of part in the rectangle (R1) is (34a,â€‰310b)_.

Calculate the area (A2) of the parabolaâ€‘2:

A=A1+A2 (8)

Substitute ab for A, and ab3 for A1 in Equation (8)

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