# Evaluate the following iterated integrals. int_0^3int_{-2}^1 (2x+3y)dxdy

Question
Integrals
Evaluate the following iterated integrals.
$$\int_0^3\int_{-2}^1 (2x+3y)dxdy$$

2021-03-03
$$\text{Given integral,}$$
$$\int_0^3\int_{-2}^1 (2x+3y)dxdy$$
$$\text{Now}$$
$$\int_0^3\int_{-2}^1 (2x+3y)dxdy=\int_0^3(\frac{2x^2}{2}+3xy)_{-2}^1 dy$$
$$=\int_0^3 (x^2+3xy)_{-2}^1 dy$$
$$=\int_0^3 ((1)^2+3(1)y-(-2)^2-3(-2)y)dy$$
$$=\int_0^3 (1+3y-4+6y)dy$$
$$=\int_0^3 (-3+9y)dy$$
$$=(-3y+\frac{9y^2}{2})_0^3$$
$$=(-3(3)+\frac{9(3)^2}{2}+3(0)+\frac{9(0)^2}{2})$$
$$=-9+\frac{81}{2}+0$$
$$=\frac{63}{2}$$
$$\text{Final answer:}$$
$$\int_0^3\int_{-2}^1 (2x+3y)dxdy=\frac{63}{2}$$

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