Question

Compute the arc length, in terms of a and b, of the curve r(θ)=−7(θ^2) from θ=a to θ=b. You may assume 0≤a≤b?

Answer 1

Step 1:

The polar curve is , .

The length of a curve with polar equation , is .

Differentiate with respect to on each side.

Substitute ,   and limits of in length of the polar curve formula. 

Consider .

Differentiate with respect to on each side.

Step 2:

Change of integral limits:

Lower limit: If , then .

Upper limit: If , then .

Substitute , and change of limits in the integral. 

Solution:

Length of the polar curve  is .

Answer 2

Step 1:

The polar curve is , .

The length of a curve with polar equation , is .

Differentiate with respect to on each side.

Substitute ,   and limits of in length of the polar curve formula. 

Consider .

Differentiate with respect to on each side.

Step 2:

Change of integral limits:

Lower limit: If , then .

Upper limit: If , then .

Substitute , and change of limits in the integral. 

Solution:

Length of the polar curve  is .