The Triple Integral of f Over the Box B

Definition: The triple integral of f over the box B is RRR Pl Pm Pn ∗ ∗ f (x, y, z) dV = liml,m,n→∞ i=1 j=1 k=1 f (x∗ijk , yijk , zijk )△V B if this limit exists. Fubini’s Theorem for Triple Integrals: If f is continuous on the rectangular box B = [a, b] X [c, d] X [r, s], then RRR Rs Rd Rb f (x, y, z) dV = r c a f (x, y, z) dx dy dz. B Exercise 1. Evaluate RRR (xy 2 + yz 3 ) dV if Q Q = { (x, y, z) : −1 ≤ x ≤ 1, 3 ≤ y ≤ 4, 0 ≤ z ≤ 2 }. Class Exercise 1. Evaluate the iterated integral. (#4,6,8) R 1 R 2x R y R 1 R 1 R √1−z2 z (a) 0 x 0 2xyz dz dy dx (b) 0 0 0 y+1 dx dz dy R √π R x R xz 2 (c) 0 x sin y dy dz dx 0 0 Definition: A solid region E is said to be of type I if it lies between the graphs of two continuous functions of x and y, that is, E = { (x, y, z) | (x, y) ∈ D, u1 (x, y) ≤ z ≤ u2 (x, y) }, where D is the projection of E onto the xy-plane. Evaluation Theorem I: Suppose that E is a type I region. Then, RR R u2 (x,y) RRR [ u1 (x,y) f (x, y, z) dz ] dA. f (x, y, z) dV = E D If the projection D of E onto the xy-plane is a type I region, then E = { (x, y, z) | a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x), u1 (x, y) ≤ z ≤ u2 (x, y) } and the equation becomes R b R g (x) R u (x,y) RRR f (x, y, z) dV = a g12(x) u12(x,y) f (x, y, z) dz dy dx E Definition A solid region E is of type II if it is of the form E = { (x, y, z) | (y, z) ∈ D, u1 (y, z) ≤ x ≤ u2 (y, z) }, where D is the projection of E onto the yz-plane. Evaluation Theorem II: Suppose that E is a type II region. Then, RRR RR R u2 (y,z) f (x, y, z) dV = [ u1 (y,z) f (x, y, z) dx ] dA. E D Definition: A solid region E is of type III if it is of the form E = { (x, y, z) | (x, z) ∈ D, u1 (x, z) ≤ y ≤ u2 (x, z) }, where D is the projection of E onto the xz-plane. Evaluation Theorem III: Suppose that E is a type III region. Then, RRR RR R u2 (x,z) f (x, y, z) dV = [ u1 (x,z) f (x, y, z) dy ] dA. E D Exercise 2. Evaluate RRR z dV , where E is the solid tetrahedron bounded by the four planes E x = 0, y = 0, z = 0, and x + y + z = 1. (Stew Sec 15.6 Ex 2) Exercise 3. Evaluate RRR √ x2 + z 2 dV , where E is the region bounded by the paraboloid E y = x2 + z 2 and the plane y = 4. (Stew Sec 15.6 Ex 3) Class 2. Evaluate the triple integral. (#10-18 even). RRRExercise (a) ez/y dV , where E = { (x, y, z) } | 0 ≤ y ≤ 1, y ≤ x ≤ 1, 0 ≤ z ≤ xy } E RRR (b) sin y dV , where E lies below the plane z = x and above the triangular region with vertices E (0,0,0), RRR (π, 0, 0), (0, π, 0) (c) xy dV , where E is bounded by the parabolic cylinders y = x2 and x = y 2 and the planes E z = RRR 0 and z = x + y (d) RRR T xyz dV , where T is the tetrahedron with vertices (0,0,0), (1,1,0), (1,0,1), and (1,0,0) (e) z dV , where E is bounded by the cylinder y 2 + z 2 = 9 and the planes x = 0, y = 3x, E z = 0 in the first octant Volume Formula: V (E) = RRR dV E Exercise 4. Find the volume V of the solid that is bounded by the cylinder y = x2 and by the planes y + z = 4 and z = 0. (Swok Sec 17.5 Ex 3) Exercise 5. Find the volume of the region Q bounded by the graphs of z = 3x2 , z = 4 − x2 , y = 0 and z + y = 6. (Swok Sec 17.5 Ex 4) Class Exercise 3. Use a triple integral to find the volume of the given solid. (#20,22) (a) The solid enclosed by the paraboloids y = x2 + z 2 and y = 8 − x2 − z 2 (b) The solid enclosed by the cylinder x2 + z 2 = 4 and the planes y = −1 and y + z = 4 Class Exercise 4. Express the integral RRR f (x, y, z) dV as an iterated integral in six different E ways, where E is the solid bounded by the given surfaces. (#30,32) (a) y 2 + z 2 = 9, x = −2, x = 2 (b) x = 2, y = 2, z = 0, x + y − 2z = 2