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Chapter 9: Q. 35 (page 756)

In Exercises 31-36 find a definite inteyral that represents the length of the specified polar curve, and then use a graphing calculator or computer algebra system to approximate the value of the integral.

  • The inner loop of the lima莽on r=1+2sin胃.

Short Answer

Expert verified

Ans:鈭�7蟺611蟺65+4sin胃d胃=2.682

Step by step solution

01

Step 1. Given information:

r=1+2sin胃

02

Step 2. The formula for finding the arc length: 

The arc length of the polar equation r=1+2sin胃with the limits 7蟺6to 11蟺6.

The formula to find the arc length of the polar curve is given by,

L=鈭�a尾f'(胃)2+(f(胃))2d胃
03

Step 3. Finding the derivatives:

Here f(胃)=r=1+2sin胃

Now differentiate f(胃)=1+2sin胃with respect to 胃.

Then f(胃)=1+2sin胃

f'(胃)=dd胃(1+2sin胃)f'(胃)=dd胃1+dd胃2sin胃

04

Step 4. Substitute values:

Then,

f'(胃)=2cos胃

Now substitute the values of f(胃),f'(胃)in L=鈭�a尾f'(胃)2+(f(胃))2d胃.

L=鈭�a尾(2cos胃)2+(1+2sin胃)2d胃

The limits are from 7蟺6to 11蟺6, then

L=鈭�7蟺011蟺6(2cos胃)2+(1+2sin胃)2d胃
05

Step 5. Simplification:

Then,

L=鈭�7蟺611蟺64cos2胃+1+4sin2胃+4sin胃d胃L=鈭�7蟺611蟺65+4sin胃d胃sincesin2胃+cos2胃=1

Then the integral 鈭�7蟺611蟺65+4sin胃d胃=2.682

06

Step 6. The required answer:

Length of the curve is equals to 2.682.

Therefore, the required answer is 鈭�7蟺611蟺65+4sin胃d胃=2.682.

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