Check the solution to "Find the iterated integral ∫01∫y2(x+2y)dxdy"

sunshine022uv

Answered question

2021-12-10

Find the iterated integral

\int_{0}^{1} \int_{y}^{2} (x + 2 y) d x d y

Bertha Jordan

Beginner2021-12-11Added 37 answers

Step 1
We have given a double integral;

\int_{0}^{1} \int_{y}^{2} (x + 2 y) d x d y

Step 2
We shall integrate this according to the order of dx and dy.

\int_{0}^{1} \int_{y}^{2} (x + 2 y) d x d y

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

= \int_{0}^{1} [\frac{2^{2} - y^{2}}{2} + 2 y (2 - y)] d y

= \int_{0}^{1} [\frac{4 - y^{2}}{2} + 4 y - 2 y^{2}] d y

= \int_{0}^{1} [2 - \frac{5 y^{2}}{2} + 4 y] d y

= {[2 y - \frac{5 y^{3}}{6} + 4 \frac{y^{2}}{2}]}_{0}^{1}

= [2 - \frac{5}{6} + 2]

= [\frac{24 - 5}{6}]

= \frac{19}{6}

Hattie Schaeffer

Beginner2021-12-12Added 37 answers

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

\Rightarrow \int_{0}^{1} 2 + 4 y - \frac{5}{2} y^{2} d y

Evaluate
Solution:

\frac{19}{6}

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