University:
Amherst College
Course:
MATH 211 | Multivariable Calculus
Academic year:

2020
Views:

317

Pages:

6
Author:

DrieniHooloki9

4. Double Integrals (from Stewart, Calculus, Chapter 15) Rd Partial Integrals: f (x, y) dy is calculated by holding x c constant and integrating with respect to y from y = c to y = d. Note that the result is a function of x. For each fixed x, the trace of f (x, y) is a curve C in the plane Rd x = constant. The partial integral A(x) = c f (x, y) dy is the area under the curve C from y = c to y = d. Rb Similarly, a f (x, y) dx is calculated by holding y constant and integrating with respect to x from x = a to x = b. The result is a function of y. Finding Volume with a Double Integral: RR f (x, y)dA is the signed volume between the surface D z = f (x, y) and the region D in the xy-plane. Iterated Integrals Vertically Simple Regions: D = {(x, y) : a  x  b, g1 (x)  y  g2 (x)} ZZ f (x, y) dA = D Z b a Z g2 (x) f (x, y) dy dx g1 (x) Horizontally Simple Regions: D = {(x, y) : c  y  d, h1 (y)  x  h2 (y)} ZZ f (x, y) dA = D Z d c Z h2 (y) f (x, y) dx dy h1 (y) Finding Area with a Double Integral: Area(D) = RR D 1 dA Ex. 1. Find the volume of the solid that lies under the surface z = xy and above the region D in the xy-plane bounded by the line y = x 1 and the parabola y 2 = 2x + 6. Ex. 2. Evaluate Z 1 0 Z 1 sin(y 2 ) dy dx. x Ex. 3. Reverse the order of integration in the integral Ex. 4. Evaluate ZZ Z 9 4 Z p y f (x, y) dx dy. 2 (x + 2y) dA, where D is the region bounded by the parabolas y = 2x2 and y = 1 + x2 . D Polar Coordinates (r, ✓) r is the signed distance from the origin ✓ is the angle measured counter-clockwise from the positive x-axis x = r cos(✓) y = r sin(✓) r 2 = x2 + y 2 tan(✓) = Double Integrals in Polar Coordinates: ZZ Z f (x, y) dA = R Ex. 5. Evaluate 2 2 ZZ ↵ Z y x b f (r cos ✓, r sin ✓) r dr d✓. a (3x + 4y 2 )dA where R is the region in the upper half plane x R 2 2 circles x + y = 1 and x + y = 4. 0 bounded by the Ex. 6. Evaluate the integral Z 0 1 Z p 1 x2 cos(x2 + y 2 ) dy dx. 0 Ex. 7. Find the volume of the region bounded by the paraboloids z = x2 + y 2 and z = 8 x2 y2 . Ex. 8. Evaluate the integrals. Z 1Z 1 (a) x3 sin(y 3 ) dy dx x2 0 (b) (c) ZZ ZZ 2 ex dA where D is the region bounded by the lines y = 2x, y = 0, and x = 1. D ex D 2 +y 2 dA where D is the top half of the disk centered at the origin of radius 3.

4 Double Integrals - Solutions

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