Evaluate the integral ∫∫R cos(x^2 + y^2 ) dA by changing to polar coordinates where R is the region that lies above the x-axis within the circle x^2 + y^2 = 9.

Question

Evaluate the integral ∫∫_R cos(x² + y² ) dA by changing to polar coordinates where R is the region that lies above the x-axis within the circle x² + y² = 9.

Answer 1

In Polar co-ordinates,

dA = rdr dθ

The region R is above x axis and within circle x² +y² =9.

It implies the integral limits are

0≤ r ≤ 3 and 0≤ θ≤ π (in polar coordinates x² +y² =r² and for half circle 0≤ θ≤ π )

So the given integral in polar co-ordinates is

∫₀^π ∫₀³ cos r² rdr dθ = ∫₀³ cos r² rdr ∫₀^π dθ = π ∫₀³ cos r² rdr

Changing variables in integration, put r² = ρ => 2rdr =dρ

This gives limit of integration, 0 to 9 for ρ.

= π/2 ∫₀⁹ cosρ dρ = π/2 [sinρ]₀⁹

=π/2 sin9