Find the signed volume between the graph of the given function and the specified rectangle. $$f ( x , y ) = x ^ { 2 } y ^ { 3 }$$ and $R = \{ ( x , y ) : 1 \leq x \leq 3 \text { and } - 1 \leq \mathrm { y } \leq 2 \}$
A) $\frac { 1 } { 8 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 65 } { 2 }$
D) $\frac { 1 } { 6 }$

Question

Quizplus · Accepted Answer

The signed volume under the graph of $f(x, y) = x^2y^3$ over the rectangle $R$ can be found by integrating the function over the given bounds. The integral is $\int_{1}^{3} \int_{-1}^{2} x^2y^3 \, dy \, dx$. Solving this double integral gives the volume as $\frac{65}{2}$.

Find the Signed Volume Between the Graph of the Given f(x,y)=x2y3f ( x , y ) = x ^ { 2 } y ^ { 3 }f(x,y)=x2y3

Find the Signed Volume Between the Graph of the Given $f ( x , y ) = x ^ { 2 } y ^ { 3 }$