Evaluate the double integral for the given function f(x, y) and the region R. f(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.

Question

Evaluate the double integral  for the given function f(x, y) and the region R. f(x, y) = 6x + 12y; R is bounded by x = 1, x = 3, y = 0 and y = x + 1.

Quizplus · Accepted Answer

To evaluate the integral, we need to first find the bounds of integration for x and y. 
From the given region, we can see that x varies from 1 to 3 and y varies from 0 to x + 1. 
Thus, we have the integral:

$\int_1^3\int_0^{x+1} (6x+12y) dydx$

Evaluating the integral, we get:

$\int_1^3\int_0^{x+1} (6x+12y) dydx = \int_1^3 [ 6xy + 6y^2 ]_0^{x+1} dx$

Simplifying, we get:

$\int_1^3 [ 6x(x+1) + 6(x+1)^2 ]dx = \frac{56}{3}$

Therefore, the answer is B.

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