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P Today s Country Radio x D WeBWork Spring2015-M x C Chegg Study Guided sol x Matthew C https: 2/Spring 2015-Math224 1-090. 15/6/? mcerbin&effectiveUser; bin&key; Nij7eNOKSB7Yvi3 M88NVGKRZ&th; &display; Mode M Q ABP E We Work Logged in as mcerbin. Log Out MATHEMATICAL ASSOCIATION 0F AMERICA K webwork spring 2015-math2241-0900 webwork15 6 MAIN MENU Courses Homework Sets webwork 15: Problem 6 webwork15 Problem 6 Prev Up Next Password/Email Grades (1 pt) consider the solid that lies above the square (in the zy-plane) R 0,1 x 0,1 and below the eliptic paraboloid z 64 -z2 +4zy-y2. 1], Problems Estimate the volume by dividing R into 9 equal squares and choosing the sample points to lie in the midpoints of each square Problem 1 Problem 2 Problem 3 Preview Answers Submit Answers Problem 4 Problem 5 You have attempted this problem 0 times. Problem 6 You have unlimited attempts remaining Problem 7 Problem 8 Email instructor Problem 9 Problem 10 Problem 11 Problem 12 Problem 13 Problem 14 Display options

Explanation / Answer

6th) To divide R into 9 equal squares, we divide the x and y axes into 3 equal parts.
?x = ?y = (1 - 0)/3.
we will get ,
(0, 1).......(1/3, 1).......(2/3, 1).....(1, 1)

(0, 2/3)....(1/3, 2/3)...(2/3, 2/3)..(1, 2/3)

(0, 1/3)....(1/3, 1/3)....(2/3, 0).....(1, 1/3)

(0, 0)........(1/3, 0).....(2/3, 0).......(1, 0)

Note that each square has area ?x ?y = 1/9.

and then to find volume we will have

midpoints yields an approximate volume of
(1/9) [f(1/6, 1/6) + f(1/6, 3/6) + f(1/6, 5/6) + f(3/6, 1/6) + f(3/6, 3/6) + f(3/6, 5/6) + f(5/6, 1/6)
+ f(5/6, 3/6) + f(5/6, 5/6)].

we will plug values for x and y in the f(x, y) = 64 -x^2 +4xy -4y^2 and all   , we will get volume .

8) As it is a rectangle, you can interchange the X and Y integral.

The way to consider it is the integrand terminals for the Y-integral are the y-bounds: y = 2 to y = 3
And likewise, for the X integral, they are the x-bounds: x = -3 to x=-2
.
Evaluating:

Inner integral becomes:
Int_1_to_4_by_y[ z ]
= [4x^2 y + 7/3 y^3] 2 to 3
= 12x^2 + 189/3 - 8x^2 - 56/3
= 4x^2 + 133/3

Outer integral is:
Int_-3_to_-2_by_x[ 4x^2 + 133/3 ]
= [4/3x^3 + 133/3x] -3 to -2
= 209/3
units ^3

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