Question

Find the arc length of the curve r=θ, for π/2≤θ≤π?

Answer 1

Step 1:

The polar curve is , .

The length of a curve with polar equation , is .

Here .

Consider .

Differentiate with respect to on each side.

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

Step 2:

Formula for integration: .

Substitute and in  the above formula.

Hyperbolic inverse sine function is defined as, .

.

Solution:

Arc length of the polar curve is

.

Answer 2

Step 1:

The polar curve is , .

The length of a curve with polar equation , is .

Here .

Consider .

Differentiate with respect to on each side.

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

Step 2:

Formula for integration: .

Substitute and in  the above formula.

Hyperbolic inverse sine function is defined as, .

.

Solution:

Arc length of the polar curve is

.