Question

Evaluate the double integral of e^(x+y) dA over the region R defined as (x, y) .

∬[0 to 8]∬[0 to 4] e^(x+y) dA = _________

Answer 1

To evaluate the double integral of e^(x+y) over the region R with bounds 0 ≤ x ≤ 8 and 0 ≤ y ≤ 4, perform sequential integration first with respect to y, then with respect to x, resulting in the solution (e^4 - 1)(e^8 - 1).

Step-by-step explanation:

The student has asked to evaluate the double integral ∫∫ e^(x+y) dA over the region R, which is defined by the bounds 0 ≤ x ≤ 8 and 0 ≤ y ≤ 4. To solve this problem, we will integrate e^(x+y) first with respect to y from 0 to 4, and then integrate the resulting expression with respect to x from 0 to 8.

First, we integrate with respect to y:

∫ e^(x+y) dy = e^x ∫ e^y dy = e^x [e^y]_0^4 = e^x (e^4 - 1)

Next, we integrate the result with respect to x:

∫ e^x (e^4 - 1) dx = (e^4 - 1) ∫ e^x dx = (e^4 - 1) [e^x]_0^8 = (e^4 - 1)(e^8 - 1)

Hence, the value of the double integral ∫∫ e^(x+y) dA over the region R is (e^4 - 1)(e^8 - 1).