5.1 Double integrals over rectangular regions  (Page 7/12)

$f(x,y)=4x+2y+8xy$

$f(x,y)=16x^2+\frac{y}{2}$

$f(x,y)=\text{sin}x-\text{cos}y,$ $R=[0,\pi ]\times [0,\pi ]$

$f(x,y)=\text{cos}x+\text{cos}y,$ $R=\left[0,\pi \right]\times \left[0,\frac{\pi }{2}\right]$

Use the midpoint rule with $m=n=2$ to estimate ${\displaystyle \underset{R}{\iint }f(x,y)dA},$ where the values of the function f on $R=[8,10]\times [9,11]$ are given in the following table.

The values of the function f on the rectangle $R=[0,2]\times [7,9]$ are given in the following table. Estimate the double integral ${\displaystyle \underset{R}{\iint }f(x,y)dA}$ by using a Riemann sum with $m=n=2.$ Select the sample points to be the upper right corners of the subsquares of R .

The depth of a children’s 4-ft by 4-ft swimming pool, measured at 1-ft intervals, is given in the following table.

1. Estimate the volume of water in the swimming pool by using a Riemann sum with $m=n=2.$ Select the sample points using the midpoint rule on $R=[0,4]\times [0,4].$
2. Find the average depth of the swimming pool.
   

   	y
   x	0	1	2	3	4
   0	1	1.5	2	2.5	3
   1	1	1.5	2	2.5	3
   2	1	1.5	1.5	2.5	3
   3	1	1	1.5	2	2.5
   4	1	1	1	1.5	2

The depth of a 3-ft by 3-ft hole in the ground, measured at 1-ft intervals, is given in the following table.

1. Estimate the volume of the hole by using a Riemann sum with $m=n=3$ and the sample points to be the upper left corners of the subsquares of R .
2. Find the average depth of the hole.
   

   	y
   x	0	1	2	3
   0	6	6.5	6.4	6
   1	6.5	7	7.5	6.5
   2	6.5	6.7	6.5	6
   3	6	6.5	5	5.6

The level curves $f(x,y)=k$ of the function f are given in the following graph, where k is a constant.

1. Apply the midpoint rule with $m=n=2$ to estimate the double integral ${\displaystyle \underset{R}{\iint }f(x,y)dA},$ where $R=[0.2,1]\times [0,0.8].$
2. Estimate the average value of the function f on R .