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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the area of the region that lies inside both curves.

34. r = a sin θ, r = b cos θ, a > 0, b > 0

To determine

To Find: The area of the region that lies inside both curves.

Explanation

Given:

The given polar equations are as below.

r=asinθ (1)

r=bcosθ (2)

Calculation:

Calculate the value of r using the equation (1).

r=asinθ

Substitute 0 for θ and 2 for a in the equation (1).

r=2sin0=0

Calculate the value of x.

x=rcosθ

Substitute 0 for r and 0 for θ .

x=0×cos(0×π180)=0

Calculate the value of y.

y=rsinθ

Substitute 0 for r and 0 for θ .

y=0×sin(0×π180)=0

Similarly calculate the values of x and y using the value of θ from 0 to 360 .

Tabulate the values of x and y in table (1).

θ r=asinθ x=rcosθ y=rsinθ
0.00 0.00 0.00 0.00
10.00 0.35 0.34 0.06
20.00 0.68 0.64 0.23
30.00 1.00 0.87 0.50
40.00 1.29 0.98 0.83
50.00 1.53 0.98 1.17
60.00 1.73 0.87 1.50
70.00 1.88 0.64 1.77
80.00 1.97 0.34 1.94
90.00 2.00 0.00 2.00
100.00 1.97 -0.34 1.94
110.00 1.88 -0.64 1.77
120.00 1.73 -0.87 1.50
130.00 1.53 -0.98 1.17
140.00 1.29 -0.98 0.83
150.00 1.00 -0.87 0.50
160.00 0.68 -0.64 0.23
170.00 0.35 -0.34 0.06
180.00 0.00 0.00 0.00
190.00 -0.35 0.34 0.06
200.00 -0.68 0.64 0.23
210.00 -1.00 0.87 0.50
220.00 -1.29 0.98 0.83
230.00 -1.53 0.98 1.17
240.00 -1.73 0.87 1.50
250.00 -1.88 0.64 1.77
260.00 -1.97 0.34 1.94
270.00 -2.00 0.00 2.00
280.00 -1.97 -0.34 1.94
290.00 -1.88 -0.64 1.77
300.00 -1.73 -0.87 1.50
310.00 -1.53 -0.98 1.17
320.00 -1.29 -0.98 0.83
330.00 -1.00 -0.87 0.50
340.00 -0.68 -0.64 0.23
350.00 -0.35 -0.34 0.06
360.00 0.00 0.00 0.00

Calculate the value of r using the equation (2).

r=bcosθ

Substitute 0 for θ and 1 for b in the equation (2).

r=1cos0=1

Calculate the value of x.

x=rcosθ

Substitute 1 for r and 0 for θ .

x=rcosθ=1×cos(0×π180)=1

Calculate the value of y.

y=rsinθ

Substitute 1 for r and 0 for θ .

y=1×sin(0×π180)=0

Similarly, calculate the values of x and y using the value of θ from 0 to 360 .

Tabulate the values of x and y in table (2).

θ r=bcosθ x=rcosθ y=rsinθ
0.00 1.00 1.00 0.00
10.00 0.98 0.97 0.17
20.00 0.94 0.88 0.32
30.00 0.87 0.75 0.43
40.00 0.77 0.59 0.49
50.00 0.64 0.41 0.49
60.00 0.50 0.25 0.43
70.00 0.34 0.12 0.32
80.00 0.17 0.03 0.17
90.00 0.00 0.00 0.00
100.00 -0.17 0.03 -0.17
110.00 -0.34 0.12 -0.32
120.00 -0.50 0.25 -0.43
130.00 -0.64 0.41 -0.49
140.00 -0.77 0.59 -0.49
150.00 -0.87 0.75 -0.43
160.00 -0.94 0.88 -0.32
170.00 -0.98 0.97 -0.17
180.00 -1.00 1.00 0.00
190.00 -0.98 0.97 0.17
200.00 -0.94 0.88 0.32
210.00 -0.87 0.75 0.43
220.00 -0.77 0.59 0.49
230.00 -0.64 0.41 0.49
240.00 -0.50 0.25 0.43
250.00 -0.34 0.12 0.32
260.00 -0.17 0.03 0.17
270.00 0.00 0.00 0.00
280.00 0.17 0.03 -0.17
290.00 0.34 0.12 -0.32
300.00 0.50 0.25 -0.43
310.00 0.64 0.41 -0.49
320.00 0.77 0.59 -0.49
330.00 0.87 0.75 -0.43
340.00 0.94 0.88 -0.32
350.00 0.98 0.97 -0.17
360.00 1.00 1.00 0.00

Graph:

The graph is plotted for x and y using the table (1) and (2) in figure (1).

Calculate the value of θ .

Equate the given polar equation (1) and (2).

asinθ=bcosθsinθcosθ=batanθ=baθ=tan1(ba)

Refer figure (1), the curves intersect at the pole when θ=0 for the sine one and θ=π2 for the cosine one.

From the points θ=0 and r=asinθ curve starts at the pole then intersects with the r=bcosθ

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P-52RECh-10 P-53RECh-10 P-54RECh-10 P-55RECh-10 P-56RECh-10 P-57RECh-10 P-58RECh-10 P-1PCh-10 P-2PCh-10 P-3PCh-10 P-4PCh-10 P-5PCh-10 P-6P

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