Evaluate the following iterated integrals. int_0^2int_0^1 4xy dx dy

Question

Evaluate the following iterated integrals.

\int_{0}^{2} \int_{0}^{1} 4 x y d x d y

Answer 1

Given integral

\int_{0}^{2} \int_{0}^{1} 4 x y d x d y

= \int_{0}^{2} 4 [x d x] y d y

= \int_{0}^{2} 4 \frac{x^{2}}{2} y d y

= \int_{0}^{2} 2 [x^{2}]_{0}^{1} y d y

= \int_{0}^{2} 2 [1^{2} - 0^{2}] y d y

= \int_{0}^{2} 2 y d y

= 2 [\frac{y^{2}}{2}]_{0}^{2}

= [y^{2}]_{0}^{2}

= 2^{2} - 0^{2}

= 4

Therefore, the result of \int_{0}^{2} \int_{0}^{1} 4 x y d x d y i s 4.

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