5. Evaluate where D is the upper solid hemisphere 2y2+ z2 < 4, z 2 0.
A solid is bounded above by a portion of the hemisphere z= 2 – – 72 . And below by the cone z = 2 + y2 , with a < 0 and y < 0. Part a: Express the volume of the solid as a triple integral involving 2, y and z. Part b: Express the volume of the solid as a triple integral in cylindrical coordinates. Parte: Express the volume of the solid as a triple integral in...
Let F = < x-eyz, xexx, z?exy >. Use Stokes' Theorem to evaluate slice curlĒ ds, where S is the hemisphere x2 + y2 + z2 = 1, 2 > 0, oriented upwards.
Let D be the solid spherical "cap" given by x2 + y2 + z2 < 16 and 2 > 1. Set up, but do not evaluate, a triple integral representing the volume of D in cylindrical coordinates.
The solid S sits below the plane z = 2x + 5 and above the region in the xy-plane where 1 < x2 + y2 = 4 and x + y < 0. The volume of S is:
in positive sense (both
exercises)
1)
2)where z2 is the root of z^3 - 1
that is in the second quadrant.
-0<R< 3 1 dz. 12-11=R 23 -1 Jos os -0907 1 dz 12-22=0.997 23 - 1
Consider the vector field F (x, y, z) = <y?, z2, x?>. Compute the curl (F). Use Stokes' Theorem to evaluate S. F. dr where C is the triangle (0,0,0), (1,0,0), and (0, 1, 1) oriented counter-clockwise when viewed from above.
2. Find the image of the upper half-plane by the mapping 1-χα 1 za where 0< α< 1 and 20 has its principal value.
mmer 2019 3. Evaluate: M y2 dV where E the solid hemisphere x2 + y2 +z2 9 and y 2 0 indrnse)
mmer 2019 3. Evaluate: M y2 dV where E the solid hemisphere x2 + y2 +z2 9 and y 2 0 indrnse)
Let z=5 where x, y, z E R. Prove that z? +z2+z?>
[3] 5. Suppose that f: D[0,1] for all z E D[0, 1] D[0,1] is holomorphic, prove that \f'(z) < 1/(1 - 121)2