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Chapter 13: Problem 5

Evaluate the following iterated integrals. $$\int_{0}^{2} \int_{0}^{1} 4 x y d x d y$$

Short Answer

Expert verified
Question: Evaluate the iterated integral of the function 4xy over the given domain: $$\int_{0}^{2} \left(\int_{0}^{1} 4 x y d x\right) d y$$ Answer: The value of the iterated integral is 4.

Step by step solution

01

Set up outer integral

Since the problem is an iterated integral, we need to start with the outer integral first. The given integral can be written as $$\int_{0}^{2} \left(\int_{0}^{1} 4 x y d x\right) d y$$ The outer integral is with respect to y and the limits of integration are from 0 to 2.
02

Compute the inner integral

Now, consider the inner integral and compute its value: $$\int_{0}^{1} 4 x y d x$$ To evaluate this integral, we treat y as a constant and integrate with respect to x. Using the power rule for integration, we have: $$\int_{0}^{1} 4 x y d x = 4 y \int_{0}^{1} x d x = 4 y \frac{1}{2} x^2 \Big|_0^1 = 4 y \frac{1}{2} (1^2 - 0^2) = 2 y$$
03

Substitute the inner integral result into the outer integral

Now that we have the result from the inner integral, substitute it into the outer integral: $$\int_{0}^{2} 2 y d y$$
04

Compute the outer integral

Finally, evaluate the outer integral by using the power rule for integration: $$\int_{0}^{2} 2 y d y = 2 \int_{0}^{2} y d y = 2 \frac{1}{2} y^2 \Big|_0^2 = y^2 \Big|_0^2 = (2^2 - 0^2) = 4$$ The value of the iterated integral is 4.

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