Question

Evaluate the integral ∬(R) e^(4x + 5y) dA, where R is the region defined as R = [0,8] × [0,1].

Answer 1

To solve the integral ∬(R) e^(4x + 5y) dA over the region R = [0,8] × [0,1], we integrate the exponential function first with respect to x and then with respect to y, utilizing the fundamental theorem of calculus.

Step-by-step explanation:

The student has asked to evaluate the double integral ∬(R) e(4x + 5y) dA, where the region R is defined by the Cartesian product of intervals [0,8] and [0,1]. To solve this, we need to integrate the function e(4x + 5y) over the domain R. We perform the integration in two steps, first with respect to x, keeping y constant, and then with respect to y.

Step-by-step, the solution process is as follows:

1. Integrate e(4x + 5y) with respect to x from 0 to 8.
2. Then, integrate the resulting expression with respect to y from 0 to 1.

The key to solving this problem is to use the fact that exponential functions are easily integrable, and since the limits of integration are constants, this problem becomes a straightforward application of the fundamental theorem of calculus.