Application of property VI of double integrals

 Property VI of double integrals (reminder)

In the case where the function f(x,y) is expressed as a product of two functions g(x)  of x only and h(y) of y only, then over the region R = {(x,y)/ a ≤ x ≤ b., c≤ y ≤ d}, the double integral of f(x, y) over the region R can be written as:

Example

Solution

It is clear that the given function f(x,y) is a product of a function of x and a function y. Here we have: g(x) = cosx and h(y) =  e^y. By substituting g(x), h(y), a and b in the equality above, we obtain:

 

Practice

a) Use the properties of double integrals and the Fubini's theorem to evaluate: