To estimate the volume of the solid using a Riemann sum, we will first divide the given rectangle R into 6 smaller rectangles (m=3 along the x-axis, n=2 along the y-axis). The dimensions of each small rectangle along the x-axis will be Δx = (14-8)/3 = 2 and along the y-axis will be Δy = (8-4)/2 = 2.
Next, we will evaluate the function z = xy at the upper right corner (sample point) of each small rectangle, and then multiply each value by the area of the rectangle (ΔxΔy = 2*2 = 4).
For the sample points, we only pick upper right corner for each small rectangle:
1. (x1, y1) = (10, 6)
2. (x2, y2) = (12, 6)
3. (x3, y3) = (14, 6)
4. (x4, y4) = (10, 8)
5. (x5, y5) = (12, 8)
6. (x6, y6) = (14, 8)
Now, let's compute the volume of each small rectangular box:
1. V1 = f(x1, y1) * ΔxΔy = (10 * 6) * 4 = 240
2. V2 = f(x2, y2) * ΔxΔy = (12 * 6) * 4 = 288
3. V3 = f(x3, y3) * ΔxΔy = (14 * 6) * 4 = 336
4. V4 = f(x4, y4) * ΔxΔy = (10 * 8) * 4 = 320
5. V5 = f(x5, y5) * ΔxΔy = (12 * 8) * 4 = 384
6. V6 = f(x6, y6) * ΔxΔy = (14 * 8) * 4 = 448
Now, adding these volumes up, we get the estimated volume of the solid as:
V = V1 + V2 + V3 + V4 + V5 + V6 = 240 + 288 + 336 + 320 + 384 + 448 = 2016
So, the estimated volume of the solid using a Riemann sum is 2016 cubic units.
(b) Using the Midpoint Rule, we will find the midpoints of each small rectangle and evaluate the function z = xy at these midpoints. Then, we will multiply the value by the area of the rectangle (ΔxΔy = 4).
Now, let's find the midpoints for each small rectangle:
1. (x1, y1) = (9, 5)
2. (x2, y2) = (11, 5)
3. (x3, y3) = (13, 5)
4. (x4, y4) = (9, 7)
5. (x5, y5) = (11, 7)
6. (x6, y6) = (13, 7)
Now, let's compute the volume of each small rectangular box using the midpoints:
1. V1 = f(x1, y1) * ΔxΔy = (9 * 5) * 4 = 180
2. V2 = f(x2, y2) * ΔxΔy = (11 * 5) * 4 = 220
3. V3 = f(x3, y3) * ΔxΔy = (13 * 5) * 4 = 260
4. V4 = f(x4, y4) * ΔxΔy = (9 * 7) * 4 = 252
5. V5 = f(x5, y5) * ΔxΔy = (11 * 7) * 4 = 308
6. V6 = f(x6, y6) * ΔxΔy = (13 * 7) * 4 = 364
Now, adding these volumes up, we get the estimated volume of the solid using the Midpoint Rule as:
V = V1 + V2 + V3 + V4 + V5 + V6 = 180 + 220 + 260 + 252 + 308 + 364 = 1584
So, the estimated volume of the solid using the Midpoint Rule is 1584 cubic units.
Estimate the volume of the solid that lies below the surface z = xy and above the following rectangle.
R =
(x, y)|8 ≤ x ≤ 14, 4 ≤ y ≤ 8
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 in part (a)
thanks so much
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