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A2) Let Sl be the unit circle z2 + y2-l in R2. Let S2 be the unit sphere z2 + y2 + z2-l in R. Let...
5. (2 points) Let S be the solid inside both x2+y2 = 16 and x2+y2 +...
5. (2 points) Let S be the solid inside both x2+y2 = 16 and x2+y2 + z2 = 32. Consider (a) Write an iterated integral for the triple integral in rectangular coordinates. (b) Write an iterated integral for the triple integral in cylindrical coordinates. (c) Write an iterated integral for the triple integral in spherical coordinates. (d) Evaluate one of the above iterated integrals.
5. (2 points) Let S be the solid inside both x2+y2 = 16 and x2+y2 +...
For the described solid S, write the triple integral f(x,y, z)dV as an iterated integral in (i) r...
For the described solid S, write the triple integral f(x,y, z)dV as an iterated integral in (i) rectangular coordinates (x,y, z); (ii) cylindrical coordinates (r, 0, 2); (iii) spherical coordinates (p, φ,0). a. Inside the sphere 2 +3+224 and above the conezV b. Inside the sphere x2 + y2 + 22-12 and above the paraboloid z 2 2 + y2. c. Inside the sphere 2,2 + y2 + z2-2 and above the surface z-(z2 + y2)1/4 d. Inside the sphere...
Q3(a) Let W be the region above the sphere x2 + y2 + z2 = 6...
Q3(a) Let W be the region above the sphere x2 + y2 + z2 = 6 and below the paraboloid z = 4 - x2 - y2 as shown in Figure Q5(a) below: Z=4-x-y? x2 + y + z = 6 Figure Q3(a) (i) Find the equation of the projection of Won the xy-plane. (ii) Compute the volume of W using polar coordinates. [16 marks] (b) Using double integral in polar coordinates, compute the following: $$*** (2x+3y) dedy [7 marks]...
1. Let E be the solid region bounded above by the sphere 4 = x2 +...
1. Let E be the solid region bounded above by the sphere 4 = x2 + y2 + z2 and below by the plane y-z =-2. a. Generate a 3D picture of the region E using 3D graphing software. b. Write the integral J [ f(x,y,z)dVas an iterated integral (in rectangular coordinates) in two different ways - one with integration with respect to z first, and one with integration with respect to y first.
(a) Let R be the solid in the first octant which is bounded above by the sphere 22 + y2+2 2 and bounded below by the cone z- r2+ y2. Sketch a diagram of intersection of the solid with the rz plane...
(a) Let R be the solid in the first octant which is bounded above by the sphere 22 + y2+2 2 and bounded below by the cone z- r2+ y2. Sketch a diagram of intersection of the solid with the rz plane (that is, the plane y 0). / 10. (b) Set up three triple integrals for the volume of the solid in part (a): one each using rectangular, cylindrical and spherical coordinates. (c) Use one of the three integrals...
Let S be the sphere r2 + y2 + z2-k2 oriented outward and let F be the vector field (r, y, 2)/(a2 +y2 +2/2. Find (i) the normal vector field n on S (ii) the normal component of F on S and (ii) the flu...
Let S be the sphere r2 + y2 + z2-k2 oriented outward and let F be the vector field (r, y, 2)/(a2 +y2 +2/2. Find (i) the normal vector field n on S (ii) the normal component of F on S and (ii) the flux of F across S
Let S be the sphere r2 + y2 + z2-k2 oriented outward and let F be the vector field (r, y, 2)/(a2 +y2 +2/2. Find (i) the normal vector field n...
5. (2 points) Let S be the solid inside both x2+y2 = 16 and x2+y2 + z2 = 32. Consider (a) Write an iterated integral for the triple integral in rectangular coordinates. (b) Write an iterated integral for the triple integral in cylindrical coordinates. (c) Write an iterated integral for the triple integral in spherical coordinates. (d) Evaluate one of the above iterated integrals.
5. (2 points) Let S be the solid inside both x2+y2 = 16 and x2+y2 +...
For the described solid S, write the triple integral f(x,y, z)dV as an iterated integral in (i) rectangular coordinates (x,y, z); (ii) cylindrical coordinates (r, 0, 2); (iii) spherical coordinates (p, φ,0). a. Inside the sphere 2 +3+224 and above the conezV b. Inside the sphere x2 + y2 + 22-12 and above the paraboloid z 2 2 + y2. c. Inside the sphere 2,2 + y2 + z2-2 and above the surface z-(z2 + y2)1/4 d. Inside the sphere...
Q3(a) Let W be the region above the sphere x2 + y2 + z2 = 6 and below the paraboloid z = 4 - x2 - y2 as shown in Figure Q5(a) below: Z=4-x-y? x2 + y + z = 6 Figure Q3(a) (i) Find the equation of the projection of Won the xy-plane. (ii) Compute the volume of W using polar coordinates. [16 marks] (b) Using double integral in polar coordinates, compute the following: $$*** (2x+3y) dedy [7 marks]...
1. Let E be the solid region bounded above by the sphere 4 = x2 + y2 + z2 and below by the plane y-z =-2. a. Generate a 3D picture of the region E using 3D graphing software. b. Write the integral J [ f(x,y,z)dVas an iterated integral (in rectangular coordinates) in two different ways - one with integration with respect to z first, and one with integration with respect to y first.
(a) Let R be the solid in the first octant which is bounded above by the sphere 22 + y2+2 2 and bounded below by the cone z- r2+ y2. Sketch a diagram of intersection of the solid with the rz plane (that is, the plane y 0). / 10. (b) Set up three triple integrals for the volume of the solid in part (a): one each using rectangular, cylindrical and spherical coordinates. (c) Use one of the three integrals...
Let S be the sphere r2 + y2 + z2-k2 oriented outward and let F be the vector field (r, y, 2)/(a2 +y2 +2/2. Find (i) the normal vector field n on S (ii) the normal component of F on S and (ii) the flux of F across S
Let S be the sphere r2 + y2 + z2-k2 oriented outward and let F be the vector field (r, y, 2)/(a2 +y2 +2/2. Find (i) the normal vector field n...
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