Q. 24 In Exercises 17-25 find a defini... [FREE SOLUTION]

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Chapter 9: Q. 24 (page 756)

In Exercises 17-25 find a definite integral expression that represents the area of the given region in the polar plane, and then find the exact value of the expression.
The region bounded by the lima莽on $r = 1 + k \sin 胃$ , where $0 < k < 1$ . Bonus: Explain why the area approaches $蟺$ as $k 鈫� 0$ .

Short Answer

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Ans:

$\lim_{k 鈫� 0} A = \frac{1}{2} \lim_{k 鈫� 0} (2 蟺 + k^{2} 蟺) = \frac{1}{2} (2 蟺 + (0)^{2} 蟺) \lim_{k 鈫� 0} A = 蟺$

Step by step solution

Step 1. Given information:

A limacon $r = 1 + k \sin 胃$

Step 2. Implying formula:

Formula to find the area is $A = {鈭�}_{伪}^{尾} \frac{1}{2} (f (胃))^{2} d 胃$ or $A = {鈭�}_{伪}^{尾} \frac{1}{2} r^{2} d 胃$ .

The limits are from 0 to $2 蟺$ .

$A = \frac{1}{2} {鈭�}_{0}^{2 蟺} (1 + k \sin 胃)^{2} d 胃$

Then,

$A = \frac{1}{2} {鈭�}_{0}^{2 蟺} (1 + k^{2} \sin^{2} 胃 + 2 k \sin 胃) d 胃 A = \frac{1}{2} {鈭�}_{0}^{2 蟺} (1 + k^{2} \frac{(1 - \cos 2 胃)}{2} + 2 k \sin 胃) d 胃 A = \frac{1}{2} {鈭�}_{0}^{2 蟺} (1 + \frac{k^{2}}{2} - \frac{\cos 2 胃}{2} + 2 k \sin 胃) d 胃$

Step 3. Continue:

Thus,

$A = \frac{1}{2} {(胃 + \frac{k^{2}}{2} 胃 - \frac{\sin 2 胃}{4} + 2 k \cos 胃)}_{0}^{2 蟺} A = \frac{1}{2} (2 蟺 + \frac{k^{2}}{2} 2 蟺 - \frac{\sin 2 路 2 蟺}{4} + 2 k \cos 2 蟺 - 2 k \cos 2 路 0) A = \frac{1}{2} (2 蟺 + \frac{k^{2}}{2} 2 蟺 - 0 + 2 k - 2 k)$

Step 4. Proving:

Then,

$A = \frac{1}{2} (2 蟺 + k^{2} 蟺)$

Therefore the area of the limacon is $A = \frac{1}{2} (2 蟺 + k^{2} 蟺)$ .

Here as $k 鈫� 0$ the area is

$\lim_{k 鈫� 0} A = \frac{1}{2} \lim_{k 鈫� 0} (2 蟺 + k^{2} 蟺) = \frac{1}{2} (2 蟺 + (0)^{2} 蟺) \lim_{k 鈫� 0} A = 蟺$

Thus when $k 鈫� 0$ the area is $蟺$ .

Hence it is proved.

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Short Answer

Step by step solution

Step 1. Given information:

Step 2. Implying formula:

Step 3. Continue:

Step 4. Proving:

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91影视

In Exercises 17-25 find a definite integral expression that represents the area of the given region in the polar plane, and then find the exact value of the expression.The region bounded by the lima莽on r=1+ksin胃, where 0<k<1. Bonus: Explain why the area approaches 蟺as k鈫�0.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Implying formula:

Step 3. Continue:

Step 4. Proving:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Geometry

Applied Mathematics

Theoretical and Mathematical Physics

Calculus

Study anywhere. Anytime. Across all devices.