(A) Estimate the volume by dividing R into 4 rectangles of equal size, each twic

Question

(A) Estimate the volume by dividing R into 4 rectangles of equal size, each twice as wide as high, and choosing the sample points to result in the largest possible Riemann sum.
Riemann sum =

(B) Estimate the volume by dividing R into 4 rectangles of equal size, each twice as wide as high, and choosing the sample points to result in the smallest possible Riemann sum.
Riemann sum =

(C) Using iterated integrals, compute the exact value of the volume.
Volume =

Consider the solid that lies above the rectangle (in the xy-plane) z = x^2 - 6y + 12 (A) Estimate the volume by dividing R into 4 rectangles of equal size, each twice as wide as high, and choosing the sample points to result in the largest possible Riemann sum. Riemann sum = (B) Estimate the volume by dividing R into 4 rectangles of equal size, each twice as wide as high, and choosing the sample points to result in the smallest possible Riemann sum. Riemann sum = (C) Using iterated integrals, compute the exact value of the volume. Volume = R = [ -2, 2] X [ 0, 2] and below the surface

Explanation / Answer

Just as we approximate area using rectangles, we can use rectangular prisms to approximate volume. Instead of a width ?x, we have a base area ?x?y. Here, ?x = ?y = 1.

For (A), we are to use z(0,0) for the height of the rectangular prism that approximates the volume above the square [0,1]*[0,1]. Similarly, we use z(0,1), z(1,0) and z(1,1) for the other heights. The lower-left-corner estimate of the volume, then, is

(1)[(100 - 0

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