1. (a) Estimate the volume of the solid that lies below the surface z = 9 x + 4

Question

1. (a) Estimate the volume of the solid that lies below the surface

z = 9x + 4y2 and above the rectangleR = [0, 2] × [0, 4]. Use a Riemann sum with m = n = 2 and choose the sample points to be lower right corners.

(b) Use the Midpoint Rule to estimate the volume in part (a).

2. If R =

[3, 1] × [0, 2],

use a Riemann sum with m = 4, n = 2 to estimate the value of

Take the sample points to be the upper left corners of the squares.

3. Estimate the volume of the solid that lies below the surface z = xy and above the following rectangle.

R =

(x, y) | 2 x 8, 8 y 12

(a) Use a Riemann sum with m = 3, n = 2, and take the sample point to be the upper right corner of each square.


(b) Use the Midpoint Rule to estimate the volume of the solid.

Explanation / Answer

Solution: (1)

A) With m = n = 2, we have x = (2 - 0)/2 = 1 and y = (4 - 0)/2 = 2.

we partition R = [0, 2] x [0, 4] into 4 squares with side lengths 2 with vertices (roughly drawn below):

(0, 4).....(1, 4)....(2, 4)
...............................
(0, 2).....(1, 2)....(2, 2)
................................
(0, 0).....(1, 0)....(2, 0)

So, using lower right endpoints yields (with f(x,y) = 9x + 4y^2):
f(x*, y*) x y
= 1 * 2 [f(1, 0) + f(2, 0) + f(1, 2) + f(2, 2)]
= 2 [9 + 18 + 25 + 34]
= 172.
------------------
B) Using the midpoints of each square instead yields
1 * 2 [f(1/2, 1) + f(1/2, 3) + f(3/2, 1) + f(3/2, 3)]
= 2 [(9/2 + 4) + (9/2 + 36) + (27/2 + 4) + (27/2 + 36)]
= 232

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.