University:
Amherst College
Course:
MATH 121 | Intermediate Calculus
Academic year:

2023
Views:

469

Pages:

10
Author:

Dustin Green

5. Triple Integrals (from Stewart, Calculus, Chapter 15) z-Simple Regions W = {(x, y) 2 D, z1 (x, y)  z  z2 (x, y)} RRR W f (x, y, z) dV = RR R z2 (x,y) D z1 (x,y) f (x, y, z) dz dA Finding Volume with a Triple Integral: Volume(W ) = RRR W 1 dV . Ex. 1. Use a triple integral to find the volume of the tetrahedron bounded by the four planes x = 0, z = 0, x = 2y, and x + 2y + z = 2. Ex. 2. Reverse the order of integration in the integral Z 1 0 Z x2 0 Z y f (x, y, z) dz dy dx. (Give all possibilities.) 0 Cylindrical Coordinates (r, ✓, z) (r, ✓) are the polar coordinates of the projection (x, y) of the point (x, y, z) in the xy-plane, with r 0 x = r cos(✓) y = r sin(✓) r 2 = x2 + y 2 tan(✓) = y x z=z Triple Integrals in Cylindrical Coordinates: ZZZ f (x, y, z) dV = W Z ✓2 ✓1 Z r2 (✓) r1 (✓) Z z2 (r,✓) f (r cos ✓, r sin ✓, z) r dz dr d✓. z1 (r,✓) Ex. 3. Find the volume of the region that is inside both the sphere x2 + y 2 + z 2 = 25 and the cylinder x2 + y 2 = 9. Spherical Coordinates (⇢, ✓, ) ⇢ 0 is the distance from the origin ✓ is the polar angle of the projection (x, y, 0) in the xy-plane is the angle of declination measured from the positive z-axis to OP x = ⇢ sin( ) cos(✓) ⇢2 = x 2 + y 2 + z 2 y = ⇢ sin( ) sin(✓) tan(✓) = y x z = ⇢ cos( ) cos( ) = z ⇢ Triple Integrals in Spherical Coordinates: ZZZ f (x, y, z) dV = W Ex. 4. Evaluate ZZZ x2 + y 2 + z 2 = 4 for z Z ✓2 ✓1 Z 2 1 Z ⇢2 (✓, ) f (⇢ sin cos ✓, ⇢ sin sin ✓, ⇢ cos ) ⇢2 sin d⇢ d d✓. ⇢1 (✓, ) z dV where W is the region above the cone x2 + y 2 = z 2 and below the sphere W 0. Ex. 5. A solid region W lies withinRRR the p cylinder x2 + y 2 = 1, below the plane z = 4, and above the 2 2 paraboloid z = 1 x y . Evaluate x2 + y 2 dV . W Ex. 6. Let F (x, y, z) = x2 + y 2 + z 2 . Compute R3 | x2 + y 2 + z 2  1}. RRR W krF k dV where W is the region {(x, y, z) 2 Ex. 7. Evaluate the integral Ex. 8. Evaluate the integral and y = 1. Z 2 2 ZZZ Ex. 9. Evaluate the integral p Z 4 x2 p 4 x2 Z 2 p (x2 + y 2 ) dz dy dx x2 +y 2 xy dV where W is the region bounded by z = 9 x2 y 2 , z = 0, y = x2 , W ZZZ z dV where W is the solid tetrahedron bounded by the four planes W x = 0, y = 0, z = 0, and x + y + z = 1. Ex. 10. Evaluate the integral ZZZ e(x W 2 +y 2 +z 2 )3/2 dV where W is the unit ball x2 + y 2 + z 2  1.

5 Triple Integrals - Solutions

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