Question

evaluate ∫ ∫ R  f(x,y) dy dx over the region

1. R= {(x,y)| 0 ≤ x ≤ 1 and 0 ≤ y ≤ x} where f(x,y)= 24y^2e^(x^4+1)

2. R={(x,y)| 0 ≤  x ≤  y and 0 ≤  y ≤  2 } where f(x,y) = 12e^(4y+2)

3. Find the volume of the solid under the graph of the function f(x,y) = 18x^2y^2 and over the region R= {(x,y)| 0 ≤ x ≤ 1 and -1 ≤  y ≤  2 }

Answer 1

Fubini's Theorem:

If is continuous on the rectangle then .

(1)

and .

Evaluate .

Answer 2

Fubini's Theorem:

If is continuous on the rectangle then .

(2)

and .

Evaluate .

Apply integration by parts.

Integration by parts: .

Consider then .

Consider then .

Answer 3

(3)

Find the volume of the solid.

and .

Evaluate .