Consider the solid that lies above the square (in the xy-plane) R=[0,2]*[0,2],
and below the elliptic paraboloid z=100−x^2−4y^2.
(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.
To solve this problem, we'll use the method of rectangular approximation to estimate the volume of the solid. We'll divide the region R into smaller squares and approximate the volume by summing the volumes of these squares.
(A) To estimate the volume by choosing the sample points in the lower left-hand corners, we'll divide R into 4 equal squares.
- Divide the side length of R (2 units) into 2 equal parts, resulting in each square having a side length of 1 unit.
- The sample points will be at the lower left-hand corners of each square, so the sample points will be (0,0), (1,0), (0,1), (1,1).
- Evaluate the height of the solid at each sample point using the equation z = 100 - x^2 - 4y^2.
- At (0,0): z = 100 - (0)^2 - 4(0)^2 = 100
- At (1,0): z = 100 - (1)^2 - 4(0)^2 = 99
- At (0,1): z = 100 - (0)^2 - 4(1)^2 = 96
- At (1,1): z = 100 - (1)^2 - 4(1)^2 = 94
- Calculate the volume of each square by multiplying the side length (1 unit) by the corresponding height.
- The volume of the square at (0,0) is 1 * 100 = 100 cubic units.
- The volume of the square at (1,0) is 1 * 99 = 99 cubic units.
- The volume of the square at (0,1) is 1 * 96 = 96 cubic units.
- The volume of the square at (1,1) is 1 * 94 = 94 cubic units.
- Sum up the volumes of these squares to estimate the total volume:
- Total volume ≈ 100 + 99 + 96 + 94 = 389 cubic units.
(B) To estimate the volume by choosing the sample points in the upper right-hand corners, follow the same process as in (A), but use the upper right-hand corners of each square as the sample points:
- The sample points will be (1,1), (2,1), (1,2), (2,2).
- Calculate the height of the solid at each sample point:
- At (1,1): z = 100 - (1)^2 - 4(1)^2 = 94
- At (2,1): z = 100 - (2)^2 - 4(1)^2 = 89
- At (1,2): z = 100 - (1)^2 - 4(2)^2 = 80
- At (2,2): z = 100 - (2)^2 - 4(2)^2 = 74
- Calculate the volume of each square and sum them up:
- The volume of the square at (1,1) is 1 * 94 = 94 cubic units.
- The volume of the square at (2,1) is 1 * 89 = 89 cubic units.
- The volume of the square at (1,2) is 1 * 80 = 80 cubic units.
- The volume of the square at (2,2) is 1 * 74 = 74 cubic units.
- Total volume ≈ 94 + 89 + 80 + 74 = 337 cubic units.
(C) To find the average of the two estimates, simply add them together and divide by 2:
- Average volume ≈ (389 + 337) / 2 = 726 / 2 = 363 cubic units.
(D) To compute the exact value of the volume using iterated integrals, we'll integrate the height function over the region R.
- Start by integrating with respect to x, treating y as a constant:
- Integrate from x = 0 to x = 2.
- ∫(100 - x^2 - 4y^2) dx = 100x - (1/3)x^3 - 4xy^2.
- Substitute the limits of integration:
- Integrate from x = 0 to x = 2: (100(2) - (1/3)(2)^3 - 4(2)y^2) - (100(0) - (1/3)(0)^3 - 4(0)y^2)
- Simplify: 200 - (8/3) - 8y^2.
- Next, integrate the resulting expression with respect to y, treating x as a constant:
- Integrate from y = 0 to y = 2.
- ∫(200 - (8/3) - 8y^2) dy = 200y - (8/3)y - (8/3)y^3.
- Substitute the limits of integration:
- Integrate from y = 0 to y = 2: (200(2) - (8/3)(2) - (8/3)(2)^3) - (200(0) - (8/3)(0) - (8/3)(0)^3)
- Simplify: 400 - (16/3) - (64/3) = 400 - (80/3) = 320/3 cubic units.
Therefore, the exact volume of the solid is 320/3 cubic units.