Let S be the surface of the solid bounded by the cylinder x ^2 + y ^2 = 9 and the double-cone z^ 2 = x ^2 + y^ 2 . Evaluate double integral <x ^3 , y^3 , cos(xy)>· dS
Let S be the surface of the solid bounded by the cylinder x ^2 + y...
4. (14 points) Using polar coordinates, set up, but DO NOT EVALUATE, a double integral to find the volume of the solid region inside the cylinder x2 +(y-1)2-1 bounded above by the surface z=e-/-/ and bounded below by the xy-plane.
4. (14 points) Using polar coordinates, set up, but DO NOT EVALUATE, a double integral to find the volume of the solid region inside the cylinder x2 +(y-1)2-1 bounded above by the surface z=e-/-/ and bounded below by the xy-plane.
9. Let Q be the solid bounded by the cylinder x2 + y2 = 1 and the planes z = 0 and z = 1 . Use the Divergence Theorem to calculate | | F . N dS where s is the surface of Q and F(x, y, z) = xi + yj + zk. (a) 67T (d) 0 (b) 1 (e) None of these (c) 3π
9. Let Q be the solid bounded by the cylinder x2 + y2...
Evaluate the Surface Integral, double integral F*ds, where F = [(e^x)cos(yz), (x^2)y, (z^2)(e^2x)] and S is a part of the cylinder 4y^2 + z^2 =4 that lies above the xy plane and between x=0 and x=2 with upward orientation (oriented in the direction of the positive z-axis). ASAP PLEASE
7. Assume (x, y,x)(2xy, y',5z - y). Let E be the solid upright cylinder between the planes z 0 and z-3 with base the disc x2 + y2 < 9, and let S be the outwardly oriented boundary surface of E. Note that S consists of three smooth surfaces; the surface Si of the cylinder, plus the top disc Di and the bottom disc D2. Follow the steps to verify the Divergence Theorem. (a) [12 pts.] Evaluate dS directly
7....
Let E be the solid that lies inside the cylinder x^2 + y^2 = 1,
above the xy-plane, and below the plane z = 1 + x. Let S be the
surface that encloses E. Note that S consists of three sides: S1 is
given by the cylinder x^2 + y^2 = 1, the bottom S2 is the disk x^2
+ y^2 ≤ 1 in the plane z = 0, and the top S3 is part of the plane z...
Let S be the union of the following: • The portion of the cylinder x ^2 + y ^2 = 4 where x ≥ 0, bounded between the planes z = 0 and z = 2. • The rectangle −2 ≤ y ≤ 2, 0 ≤ z ≤ 2 in the yz-plane. Evaluate the integral Z Z S xz dS
Let S be the union of the following: • The portion of the cylinder x ^2 + y ^2 = 4 where x ≥ 0, bounded between the planes z = 0 and z = 2. • The rectangle −2 ≤ y ≤ 2, 0 ≤ z ≤ 2 in the yz-plane. Evaluate the integral Z Z S xz dS
(1) Let P denote the solid bounded by the surface of the hemisphere z -Vl-r-y? and the cone2y2 and let n denote an outwardly directed unit normal vector. Define the vector field F(x, y, z) = yi + zVJ + 21k. (a) Evaluate the surface integral F n dS directly without using Gauss' Divergence Theorem. (b) Evaluate the triple integral Ш div(F) dV directly without using Gauss' Diver- gence Theorem Note: You should obtain the same answer in (a) and...
Let Surface S be that portion of the cylinder x2 + y2 = 9, which lies between the planes z = y and z = 6. a.) Sketch the Surface S. b.) Parametrize the Surface S. c.) Evaluate the following Surface Integral: ∫∫(y-z)dS
Question 3. Let Q be the solid hemisphere bounded by x + y² + 2 = 1 for 2 > 0 and by the plane z = 0, and let F = xi+yi + zk be a vector field. Verify the divergence theorem for Q and F by answering parts (a) and (b) below. Part (a) (5 points). Find the value of the triple integral of the divergence of F over the solid hemisphere Q. Part (b) (10 points). Evaluate...