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

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

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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