We have an answer to "Evaluate the integral ∫∫R(x−y2)dxdy over the region R={(x,y)|2≤x≤3,1≤y≤2}"

arenceabigns

Answered question

2021-10-13

Evaluate the integral

\int \int_{R} (x - y^{2}) d x d y

over the region R={(x,y)|

2 \leq x \leq 3, 1 \leq y \leq 2

}

Pohanginah

Skilled2021-10-14Added 96 answers

Step 1
Given integral is

\int \int_{R} (x - y^{2}) d x d y

where
R={(x,y)|

2 \leq x \leq 3, 1 \leq y \leq 2

}
Step 2
We evaluate the integral as follows.

\int \int_{R} (x - y^{2}) d x d y = \int_{2}^{3} \int_{1}^{2} (x - y^{2}) d y d x = \int_{2}^{3} {[x y - \frac{y^{3}}{3}]}_{1}^{2} d x

= \int_{2}^{3} [(2 x - \frac{8}{3}) - (x - \frac{1}{3})] d x

= \int_{2}^{3} [x - \frac{7}{3}] d x

= {[\frac{x^{2}}{2} - \frac{7}{3} x]}_{2}^{3} = (\frac{9}{2} - 7) - (2 - \frac{14}{3}) = \frac{9}{2} - 5 + \frac{14}{3} = \frac{25}{6}

Step 3
Ans:

\int \int_{R} (x - y^{2}) d x d y = \frac{25}{6}

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