Math 2263 Multivariable Calculus
Homework 12: 15.1 #4,12
June 30, 2011
15.1#4
(a) Estimate the volume of the solid that lies below the surface z = x + 2y 2 and above the
rectangle R = [0, π] × [0, π]. Take the sample points to be lower left corners.
V ≈ [f (1, 2) + f (2, 2) + f (1, 0) + f (2, 0)]∆A
= [(1 + 8) + (2 + 8) + (1 + 0) + (2 + 0)](2)
= [9 + 10 + 1 + 2](2) = 44
(b) Use the Midpoint Rule to estimate the integral in part (a).
V ≈ [f (1/2, 3) + f (1/2, 1) + f (3/2, 3) + f (3/2, 1)]∆A
= [(1/2 + 18) + (1/2 + 2) + (3/2 + 18) + (3/2 + 2)](2)
= [1 + 20 + 3 + 20](2) = 88
15.1#12
Evaluate the double integral by identifying it as the volume of a solid.
ZZ
(5 − x) dA, R = {(x, y)|0 ≤ x ≤ 5, 0 ≤ y ≤ 3}
R
This integral can be identified with the volume of the triangular prism with vertices
at (0, 0, 0), (5, 0, 0), (0, 3, 0), (5, 3, 0), (0, 0, 5), and (0, 3, 5).
volume = area of triangle × height
75
1
= (5)(5) × 3 =
2
2