By Fubini's Theorem, the double integral $\iint$R $\left( 2 x + 3 y ^ { 2 } \right) d A$ with $R = \{ ( x , y ) : 0 \leq x \leq 2,0 \leq y \leq 3 \}$ is

A)64
B)66
C)72
D)76
E)84

Question

By Fubini's Theorem, the double integral $\iint$R $\left( 2 x + 3 y ^ { 2 } \right) d A$ with $R = \{ ( x , y ) : 0 \leq x \leq 2,0 \leq y \leq 3 \}$ is

A)64
B)66
C)72
D)76
E)84

Quizplus · Accepted Answer

Fubini's Theorem allows us to compute the double integral by iteratively integrating one variable at a time. The integral becomes $\int_{0}^{2} \int_{0}^{3} (2x + 3y^2) dy dx$. First, integrate with respect to $y$, treating $x$ as a constant, to get $\int_{0}^{2} [2xy + y^3]_{0}^{3} dx$, which simplifies to $\int_{0}^{2} (6x + 27) dx$. Then, integrate with respect to $x$ to get $[3x^2 + 27x]_{0}^{2}$, which evaluates to $12 + 54 = 66$.

By Fubini's Theorem, the Double Integral ∬\iint∬ R (2x+3y2)dA\left( 2 x + 3 y ^ { 2 } \right) d A(2x+3y2)dA

By Fubini's Theorem, the Double Integral $\iint$ R $\left( 2 x + 3 y ^ { 2 } \right) d A$