Evaluate the following iterated integrals. int_1^3int_1^2(y^2+y)dx dy

Question
Integrals
asked 2021-01-28
Evaluate the following iterated integrals. \(\int_1^3\int_1^2(y^2+y)dx\ dy\)

Answers (1)

2021-01-29
\(\text{Let the given integral be I -}\)
\(I=\int_1^3(y^2+y)[\int_1^2 dx]dy\)
\(\text{Integrating with respect to x,}\)
\(I=\int_1^3(y^2+y)[x]_1^2 dy\)
\(I=\int_1^3(y^2+y)(2-1)dy\)
\(I=\int_1^3(y^2+y)dy\)
\(\text{Integrating equation with respect to y,}\)
\(I=\int_1^3(y^2+y)dy\)
\(I=[\frac{y^3}{3}+\frac{y^2}{2}]_1^3\)
\(I=(\frac{3^3}{3}+\frac{3^2}{2})-(\frac{1^3}{3}+\frac{1^2}{2})\)
\(I=\frac{27}{3}+\frac{9}{2}-\frac{1}{3}-\frac{1}{2}\)
\(I=\frac{54+27-2-3}{6}\)
\(I=\frac{76}{6}\)
\(I=\frac{38}{3}\)
\(\text{Hence,}\)
\(\int_1^3\int_1^2 (y^2+y)dxdy=\frac{38}{3}\)
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