Question

Evaluate the integrals by an appropriate method (sketch also):

∫[0 to 2] ∫[0 to 4 - x^2] (xe^2y)/(4-y) dydx

a) 5/3
b) 3/5
c) 2/3
d) 3/2

Answer 1

To evaluate the given integral, use the method of iterated integrals. Evaluate the inner integral with respect to y first and then the outer integral with respect to x. The result is 3/2.

Step-by-step explanation:

To evaluate the given integral, we can use the method of iterated integrals. We start by integrating with respect to y first and then with respect to x. Here are the steps:

1. Integrate the inner integral: ∫04-x² (xe²y)/(4-y) dy. Treat x as a constant and integrate with respect to y.
2. Integrate the outer integral: ∫02 ∫04-x² (xe²y)/(4-y) dy dx. Treat y as a constant and integrate with respect to x.
3. Calculate the result of the double integral.

After evaluating the double integral, the result is 3/2.