Question

Using polar coordinates, evaluate the integral which gives the area that lies in the first quadrant between the circles (x²+y²=196) and (x²-14x+y²=0)?

Answer 1

To evaluate the integral that gives the area between the two circles (x²+y²=196) and (x²-14x+y²=0) using polar coordinates, we find the points of intersection of the two circles in the first quadrant and then calculate the integral for the area.

Step-by-step explanation:

To evaluate the integral that gives the area between the two circles using polar coordinates, we need to find the points of intersection of the two circles in the first quadrant. To do this, we set both equations equal to each other:

x^2 + y^2 = 196

x^2 - 14x + y^2 = 0

Solving these equations, we find the points of intersection to be (7, 5) and (9, 3). The integral for the area is given by:

A = ∫(1/2)r^2 dθ = ∫(1/2)(196)r^2 dθ

Integrating from θ = 0 to θ = π/4 (since we only want the area in the first quadrant), we have:

A = (1/2)(196)(π/4) = 98π/4 = 24.5π