Estimate the volume of the solid that lies below the surface z = xy and above the following rectangle. R= -{(x, y) 13 ≤x≤ 9,1 SY55} Exercise (a) Use a Riemann sum with m = 3, n = 2, and take the sample point to be the upper right corner of each square. Step 1 We are requested to use m = 3 and n = 2 over the rectangle R = y = {(x, y) 13 ≤ x ≤ 9,1 ≤ y ≤ yss} The subrectangles are, ther 5
Estimate the volume of the solid that lies below the surface z = xy and above the following rectangle. R= -{(x, y) 13 ≤x≤ 9,1 SY55} Exercise (a) Use a Riemann sum with m = 3, n = 2, and take the sample point to be the upper right corner of each square. Step 1 We are requested to use m = 3 and n = 2 over the rectangle R = y = {(x, y) 13 ≤ x ≤ 9,1 ≤ y ≤ yss} The subrectangles are, ther 5
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Estimate the volume of the solid that lies below the surface z = xy and above the following rectangle.
R =
-{(x, y) 13 ≤x≤ 9,1 SY55}
Exercise (a)
Use a Riemann sum with m = 3, n = 2, and take the sample point to be the upper right corner of each square.
Step 1
We are requested to use m = 3 and n = 2 over the rectangle R = {(x, y) | 3 ≤ x ≤ 9, 1 ≤ y ≤ 5}.
y
5
(3,1)
S
Q Seych
The subrectangles are, therefore, as follows.
Transcribed Image Text:(3, 1)
5
Submit
7
We can estimate the volume of the given solid using the Riemann sum
Since 3 ≤ x ≤ 9 and n = 3, then Ax = 2 ✓
A4 = 4√
9
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i=1j = 1
f(x, y)44, with f(x, y) = xy.
2, and since 1 ≤ y ≤ 5 and n = 2, then Ay = 2✔
Step 2
When i = 1 and j = 1, we are referring to the coordinates of the rectangle in the lower left corner of the grid. We are told to use the upper right corner as a sample point.
1.1
Therefore, for i = 1 and j = 1, the sample point would be (x, y) =
Exercise (b)
Use the Midpoint Rule to estimate the volume of the solid.
2. Consequently, we have
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