Calculate the following integral exactly.(Your answer should not be a decimal approximating the true answer, but should be exactly equal to the true answer.Your answer may contain e, $\pi$, $\sqrt { 2 }$ , and so on.) $\int _ { 3 } ^ { 4 } \int _ { 0 } ^ { y } y ^ { 2 } e ^ { x y } d x d y$
A) $\frac { 1 } { 2 } \left( - 7 - e ^ { 9 } + e ^ { 16 } \right)$
B) $\frac { 1 } { 2 } \left( 25 - e ^ { 9 } + e ^ { 16 } \right)$
C) $\frac { 1 } { 2 } \left( - 7 + e ^ { 9 } - e ^ { 16 } \right)$
D) $\frac { 1 } { 3 } \left( - 7 - e ^ { 9 } + e ^ { 16 } \right)$

Question

Calculate the following integral exactly.(Your answer should not be a decimal approximating the true answer, but should be exactly equal to the true answer.Your answer may contain e,  $\pi$, $\sqrt { 2 }$ , and so on.) $\int _ { 3 } ^ { 4 } \int _ { 0 } ^ { y } y ^ { 2 } e ^ { x y } d x d y$
A) $\frac { 1 } { 2 } \left( - 7 - e ^ { 9 } + e ^ { 16 } \right)$
B) $\frac { 1 } { 2 } \left( 25 - e ^ { 9 } + e ^ { 16 } \right)$
C) $\frac { 1 } { 2 } \left( - 7 + e ^ { 9 } - e ^ { 16 } \right)$
D) $\frac { 1 } { 3 } \left( - 7 - e ^ { 9 } + e ^ { 16 } \right)$

Quizplus · Accepted Answer

To solve the given double integral $\int _ { 3 } ^ { 4 } \int _ { 0 } ^ { y } y ^ { 2 } e ^ { x y } d x d y$, we first integrate with respect to $x$, treating $y$ as a constant. The integral of $y^2e^{xy}$ with respect to $x$ is $\frac{y^2}{y}e^{xy} = ye^{xy}$, evaluated from $0$ to $y$. This gives us $ye^{y^2} - y$, since $e^{0} = 1$.Next, we integrate $ye^{y^2} - y$ with respect to $y$ from $3$ to $4$. The integral of $ye^{y^2}$ requires a substitution ($u = y^2$, $du = 2y dy$), leading to $\frac{1}{2}e^{y^2}$, and the integral of $y$ is $\frac{1}{2}y^2$. Evaluating these from $3$ to $4$ gives $\frac{1}{2}(e^{16} - e^9) - \frac{1}{2}(16 - 9)$, which simplifies to $\frac{1}{2}(-7 - e^9 + e^{16})$.

Calculate the Following Integral Exactly π\piπ 2\sqrt { 2 }2​ , and So On

Calculate the Following Integral Exactly $\pi$ $\sqrt { 2 }$ , and So On