Multiple Integrals

1. Estimate the volume of the solid that lies below the surface $z=xy$ and above the rectangle

   $R=\left\{\left(x,y\right)|0\le x\le 6,0\le y\le 4\right\}$.

   Use a Riemann sum with $m=3$ and $n=2$, and take the sample point to be the upper right corner of each subrectangle. Then use the Midpoint Rule to estimate the volume of the same solid.

   $288$ and $144$.

2. Let $V$ be the volume of the solid that lies under the graph of

   $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=\sqrt{52-x^2-y^2}$

   and above the rectangle given by $2\le x\le 4$ and $2\le y\le 6$. Use the lines $x=3$ and $y=4$ to divide $R$ into subrectangles. Let $L$ and $U$ be the Riemann sums computed in lower left and upper right corners respectively. Without calculating numbers $V$, $L$, and $U$, arrange them in increasing order and explain your reasoning.

   $U<V<L$

3. The figure shows level curves of a function $f$ in the square $R=\left[0,2\right]\times \left[0,2\right]$. Use the Midpoint Rule with $m=n=2$ to estimate ${\displaystyle \underset{R}{\iint }f<!--FUNCTION\; APPLICATION-->\left(x,y\right)dA}$.

   

4. Evaluate the double integral

   ${\displaystyle \underset{R}{\iint }\left(5-x\right)dA}$

   over $R=\left\{\left(x,y\right)|0\le x\le 5,0\le y\le 3\right\}$.

5. If $R=\left[0,1\right]\times \left[0,1\right]$, show that

   ${\displaystyle 0\le \underset{R}{\iint }\sin <!--FUNCTION\; APPLICATION-->\left(x+y\right)dA\le 1}$.