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Consider the solid that lies above the square and below the elliptic paraboloid

ID: 2841313 • Letter: C

Question

Consider the solid that lies above the square and below the elliptic paraboloid .

(A) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the lower left hand corners.

(B) Estimate the volume by dividing R into 4 equal squares and choosing the sample points to lie in the upper right hand corners..

(C) What is the average of the two answers from (A) and (B)?

(D) Using iterated integrals, compute the exact value of the volume.

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Consider the solid that lies above the rectangle (in the xy-plane) ,
and below the surface .

(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 =

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Evaluate the iterated integral

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Evaluate the double integral  where  is the triangular region with vertices .

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Let D be the region in the x,y-plane enclosed by the curves  and . Find the volume of the solid with base D that lies under the surface .
Volume =

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Find the volume of the solid bounded by the planes x = 0, y = 0, z = 0, and x + y + z = 2.

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Suppose  is the shaded region in the figure, and  is a continuous function on . Find the limits of integration for the following iterated integral.

(a)

A = B = C = D =

Explanation / Answer

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