Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between?

In polar coordinates:

x^2 + y^2 = 196 = 14^2 ==> r = 14.

x^2 - 14x + y^2 = 0 ==> r^2 - 14r cos θ = 0 ==> r = 14 cos θ.

Note that the second circle is inside the first circle.

So, the area ∫∫ 1 dA equals

∫(θ = 0 to π/2) ∫(r = 14 cos θ to 14) 1 * (r dr dθ)

= ∫(θ = 0 to π/2) (1/2)r^2 {for r = 14 cos θ to 14} dθ

= ∫(θ = 0 to π/2) 98(1 - cos^2(θ)) dθ

= ∫(θ = 0 to π/2) 98 sin^2(θ) dθ

= ∫(θ = 0 to π/2) 98 * (1/2)(1 - cos(2θ)) dθ

= 49(θ - sin(2θ)/2) {for θ = 0 to π/2}

= 49π/2.

I hope this helps!