Question

find the area of the region that lies inside both r=sin(θ) and r=cos(θ). hint: the final example on the final video lecture goes through a similar problem.

Answer 1

Okay, let's solve this step-by-step:

1) The equations for the two curves are:

r = sin(θ) and r = cos(θ)

2) We need to find the intersection points of these two curves. This is done by setting them equal and solving for θ:

sin(θ) = cos(θ)

=> θ = π/4

3) The intersection points are (1, π/4) and (1, 3π/4). The region lies between θ = π/4 and θ = 3π/4.

4) To find the area, we use the formula:

A = ∫θ=3π/4 θ=π/4 2πr dθ

5) Substitute r = sin(θ) or r = cos(θ):

A = ∫θ=3π/4 θ=π/4 2πsin(θ) dθ

= 2π ∫θ=3π/4 θ=π/4 sin(θ) dθ

6) Integrate:

A = 2π(cos(θ) - sin(θ) )|π/4 to 3π/4

= 2π(0 - 1) = 2π

7) Therefore, the area of the region is 2π square units.

Let me know if you have any other questions!