Write a triple integral which computes the integral of the function f(x,y,z) = x2 over the region R which is (0:1): x2+ y2 =y, x20, 1) inside the half-cylinder of radius 1/2 and centered at 0, 2) bounded below by the half-sphere z= V10 - x2 - y2, 3) bounded above by the plane z = v10

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Write a triple integral which computes the integral of the function f(x,y,z) = x2 over the region R which is (0:1): x2+ y2 =y, x20, 1) inside the half-cylinder of radius 1/2 and centered at 0, 2) bounded below by the half-sphere z= V10 - x2 - y2, 3) bounded above by the plane z = v10. Write this as a triple integral in cylindrical coordinates 2 y T Let S denote the portion of the surface z=x2 +4 above the triangle on the xy plane with vertices (0,0) (0,8), and (8,8). a) Write a parametrization of this surface in terms of "x", "y" r(x,y)= (0,00 b) Write a double integral in terms of the coordinates "x", "y", which computes the value of the integra SS_21/4+9x do falso denoted sometimes S52274 214+ 9x ds ] Simplify the integrand as much as possible 8 y2 0 y1 S542/4+9X do = S59x9) dy dx YES g(x,y)= Find the point of intersection of the line with equation L: (x,y,z) = (1, -1,0) + t(1,1,2) and the plane with equation 5x + 5y + z = 5 Point= ((,00 graph of the level curves of f(x,y) = 4x2 + 4y2- 1 z=20 174 1 N z=8 2 z=20 1 1/4 1 z=8 2 z=16 1 1 2 Z=4 Consider the surface S consisting of the portion of the graph of the function z=> = x2 + y2 above the annulus 100 < x2 + y2 s 121. Choose the normal vector n to the surface which has positive third component. A) Give a parametrization of Sin terms of the variables r, from cylindrical coordinates r(r,0) =(ODD) X B) Consider the vector field F = 1 2 + y2 -,VZ). x2 + y2 Write a double integral in terms of r, which computes the flux of F across S: 02 r2 Flux= SSO dr de 01 r1 01 = 02 = r1= r2= Write a triple integral which computes the volume of the region R in the first octant bounded by the planes x = 0, y = 0,2 = 0 and the graph of the function z = 16 - x2 - y [see figure below). Write the triple integral in the order of integration dy dx dz. 22 x2 y2 SSS dy dx dz z1 x1 y1 2 2 fis A) Suppose F(x,y)= M(x,y) i + N(x,y) j is a vector field on the xy plane such that Sc, F-dr=2 SoFode=-1 Sofa-1 where the curves C1,C2,C3 are oriented as shown in the figure below. If R denotes the interior of the triangle shown in the figure, then ON ?? SSRE dA = Rox ?? B) Consider a second vector field G(x,y)= 2 N(x,y) i - 2 M(x,y); where N M are the same functions used to define F(x,y). Then the outward flux of G across the triangle shown in the figure shown below equals outward flux = C3 C2 R C Find the values of "a" such that f(x,t) = e ax +8t satisfies the equation of 22f otox2 smallest value of "a" = a largest value of "a"= Consider the surface S consisting of the part of the paraboloid z=9 – x2 - y2 in the first octant. Let C denote the boundary curve of S, oriented counterclockwise when viewed from the positive z-axis. a) Write a parametrization of the surface in terms of the coordinates "x,y" of cartesian coordinates or the coordinates "r,0" from cylindrical coordinates. r(x,y) [or r(r,0)] = (0, b) Find the formula for or or or dr X ???? (alternatively the vector = x ar X 9) or ?? ? ?? dr or X or ?? S Consider the vector field F(x,y,z)=(yz, – xz,1). c) Find the curl of F curl F(x,y,z) = (0,00) d) Use Stokes' theorem to write a double integral in polar coordinates which computes the work (circulation) of F along the curve C. Do not find the integral. For the integrand, enter the coefficients in front of the appropriate power or "r". The coefficients can be any number, including one or zero, or negative. 02 r2 Sof_dra SS02+0R+r) dr do • 01 r1 01 = 02= r1 = r2 = Consider the surface S consisting of the part of the cone z = 2 - - 1x2 + y2 above the plane z = 0, together with the disk x2 + y2 54. Suppose we are interested in finding the flux of the vector field F= ,0 across the surface S using the normal vector n pointing away from the origin. 3'3 In order to find it, we would like to apply the divergence theorem to an appropriate three-dimensional region. a) Divergence of F(x,y,z) = = b) Write (but do not compute!) a triple integral in cylindrical coordinates which gives the flux of F 02 r2 z2 Flux= SSS dz dr de 01 r1 z 1 01 = 02 = r1= r2= 21= 22= Using the Lagrange multipliers method, find the distance of the plane 3x + 3y + 9z = 10 to the point P = (0,0,0). a) Consider the function to minimize as f(x,y,z) = x2 + y2 + z2, which is the distance squared from a point (x,y,z) on the plane to the origin (0,0,0). Set up the system of equations which you must solve (the coefficients you must enter can be any number, including zero or one) 2x - y+ z = 32 ZE y+ 3x + 3y + z = 10 b) Find the value of the Lagrange multiplier à: 2= c) Find the value of the distance of the plane to the origin distance = d) Use Stokes' theorem to write a double integral in polar coordinates which computes the work (circulation) of F along the curve C. Do not find the integral. For the integrand, enter the coefficients in front of the appropriate power or "r". The coefficients can be any number, including one or zero, or negative. 02 r2 S.F-dr= SS. SSOR+) | 2 + r) dr de 01 r1 01 = 02 = r1 = r2 =