Estimate the volume v of the mound by computing the average

Question

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 sums_4,3 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].