Then the double integral in polar coordinates is given by the formula ∬ R f (x,y)dxdy = β ∫ α h(θ) ∫ g(θ) f (rcosθ,rsinθ)rdrdθ The region of integration (Figure 3) is called the polar rectangle if it satisfies the following conditions 0 ≤ a ≤ r ≤ b, α ≤ θ ≤ β, where β−α ≤ 2πDouble integrals in Cartesian coordinates (Section 152) Start with the outer limits x ∈ 0,3 y 6 2 − 2x/3 and y > 2 r 1 − x2 32 The lower limit is part of the ellipse x2 32 y2 22 = 1 x2 2 y 3 Double integrals in Cartesian coordinates (Section 152)As argued above, the original integral will be twice as large as this integral It follows that ∫ 1 0 ∫ 1 0 emax(x2,y2)dydx= e−1 ∫ 0 1 ∫ 0 1 e max ( x 2, y 2) d y d x = e − 1
2
