1. Find the minimum and maximum values of the function subject to the constraint.
F(x. y, z) = 3x + 2y + 4z, x2 + 2y2 + = 1
2. Compute the Riemann sum S for the double integral where R = [1, 4] x [1, 3], for the grid and sample points shown in figure below.
3. Evaluate ∫∫R (50 - 10x) dA, where R = [0, 5] x [0, 3].
4. The following table gives the approximate height (in meters) at quarter-meter intervals of a mound of gravel.
|
0.75
|
0.2
|
0.5
|
0.5
|
0.25
|
0.2
|
|
0.5
|
0.5
|
0.6
|
0.8
|
0.7
|
0.5
|
|
0.25
|
0.05
|
0.5
|
0.7
|
0.6
|
0.5
|
|
0
|
0.2
|
0.25
|
0.5
|
0.25
|
0.2
|
|
y/x
|
0
|
0.25
|
0.5
|
0.75
|
1
|
Estimate the volume V of the mound by computing the average of the two Riemann sums4,3 with lower-left and upper-right vertices of the sub rectangles as sample points.
5. Use symmetry to evaluate the double integral.
∫∫R 5 sin dA, R = [0, 2π] x [0, 2π].