Question

Find the volume of the region bounded above by the surface z=4 cos xsin y and below by the rectangle R: OSXS Rosxs osys VE (Simplify your answer. Type an exact answer, using radicals as needed. Type your answer in factored form. Use integers or fractions for any numbers in the expression)

Answer 1

2

Explanation:

The region bounded above by the surface z=4 cos xsin y and below by the rectangle R:

We can use a double integral to find the volume of the region:

V = ∫∫R 4cos(x)sin(y) dA

where R is the rectangle defined by:

0 ≤ x ≤ π/2

0 ≤ y ≤ π/2

Then we can evaluate the integral as follows:

V = ∫∫R 4cos(x)sin(y) dA

= ∫0^(π/2) ∫0^(π/2) 4cos(x)sin(y) dxdy

= ∫0^(π/2) [4sin(x)](0 to π/2) dy

= ∫0^(π/2) 4sin(π/2) dy

= 4(sin(π/2))(π/2 - 0)

= 4(1)(π/2)

= 2π

Therefore, the volume of the region bounded above by the surface z=4 cos xsin y and below by the rectangle R is 2π.