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Use cylindrical coordinates to work out the volume of a ball of radius 1, and to find the center of mass of the upper half of of the ball. (If you take the hemisphere to have its origin at (0,0,0...
Use rectangular, cylindrical and spherical coordinates to set up the triple integrals representing the volume of the region bounded below by the xy plane, bounded above by the sphere with radius and...
Use rectangular, cylindrical and spherical coordinates to set up the triple integrals representing the volume of the region bounded below by the xy plane, bounded above by the sphere with radius and centered at the origin the equation of the sphere is x2 + y2 + z2-R2), and outside the cylinder with the equation (x - 1)2 +y2-1 (5 pts each) Find the volume by solving one of the triple integrals from above.( 5 pts) Total of 20 pts)
Use...
QUESTION 9 A cake is shaped like a hemisphere of radius 4 units with its base on the xy plane (a) Find the volume of th...
QUESTION 9 A cake is shaped like a hemisphere of radius 4 units with its base on the xy plane (a) Find the volume of the cake using spherical coordinates (5 marks) (b) Now suppose the cake is sliced by a plane perpendicular to the xy -plane at x = a, a > 0 . Let D be the smaller of two pieces produced. Set up a suitable integral for the volume of D (DO NOT EVALUATE). (7 marks)
QUESTION...
1. Use cylindrical coordinates to SET UP the integral for the volume of the portion of...
1. Use cylindrical coordinates to SET UP the integral for the volume of the portion of the unit ball, 22 +232 + x2 < 1, above the plane z = 12 2. (a) Write in spherical coordinates the equations of the following surfaces: (i) x2 + y2 + x2 = 4 (ii) z = 3x2 + 3y2 (b) SET UP the integral in spherical coordinates for the volume of the solid inside the surface 22 + y2 + x2 =...
Use cylindrical coordinates to find the volume of the cone shown in the figure:
Use cylindrical coordinates to find the volume of the cone shown in the figure: a) [3 points] Set up integral to find the volume b) [2 points] Evaluate the integralc) [5 points] Show that z = h(1-r/r0), where r is a radius of a circular cross-section of the cone parallel to xy-plane.
3. 3D stuff. cylindrical coordinates. A cone of uniform mass density Po has its tip at...
3. 3D stuff. cylindrical coordinates. A cone of uniform mass density Po has its tip at the origin and its axis of symmetry is aligned with the z axis. The base of the cone is at H and has radius R. Draw a big picture! Compute the following things a. the total mass of the cone. b. the center of mass of the cone. c. its moment of inertia I2z around the z axis
11. Evaluate S. 'S*(1 + 3x2 + 2y?) dx dy. 12. Find the volume in the...
11. Evaluate S. 'S*(1 + 3x2 + 2y?) dx dy. 12. Find the volume in the first octant of the solid bounded by the cylinder y2 + z2 = 4 and the plane x = 2y. Graph for Problem 12 13. Find the volume under the paraboloid z = 4 - x2 - y2 and above the xy-plane. N Consider the solid region bounded above by the sphere x + y + z = 8 and bounded below by the...
1. Using polar coordinates in the x-y plane, find the volume of the solid above the cone z r and below the hemisphe...
1. Using polar coordinates in the x-y plane, find the volume of the solid above the cone z r and below the hemisphere z= v8-r2. As a check the answer is approximately 13.88 but of course you have to calculate the exact answer 2. At the right is the graph of the 8-leafed rose r 1+2cos(40) Calculate the area of the small leaf. As a check the answer is 0.136 to 3 places of decimal (But of course you have...
Question 2 (1 point) Identify the surface r = 1, in cylindrical coordinates. Plane Cone Half...
Question 2 (1 point) Identify the surface r = 1, in cylindrical coordinates. Plane Cone Half plane Disc Sphere Circle Line segment Cylinder Use spherical coordinates to find the volume of the solid that lies above the cone z = V3x2 + 3y2 and below the sphere x2 + y2 + 2? first octant. Write = 1 in the V = L*S*%' * sin ødpdepdo 1. O 2. 1 d = < 3. À b= 4. 7T 2 5. Ő...
GIVEN: Ω isthe portion of the surface of the sphere centered at the origin of radius 3 above 1.2 1(xy, z) the plane, z-2: Ω: the field F = (x, x,x). a) FIND the flux of VrF through Ω in the given...
GIVEN: Ω isthe portion of the surface of the sphere centered at the origin of radius 3 above 1.2 1(xy, z) the plane, z-2: Ω: the field F = (x, x,x). a) FIND the flux of VrF through Ω in the given direction: n has positive 2-component. HINT: (radius a)on Q:(spherical coordinates) b) Parameterize the path,c-a2, (r,g,z)asin g dode with orientation to agree with the given n for Ω ANS: (a) 5 c) With positive orientation,an -e DETERMINE: F.ds ANS:...
NO.25 in 16.7 and NO.12 in 16.9 please. For the vector fied than the vecto and outgoing arrows. Her can use the formula for F to confirm t n rigtppors that the veciors that end near P, are shorter rs...
NO.25 in 16.7 and NO.12 in
16.9 please.
For the vector fied than the vecto and outgoing arrows. Her can use the formula for F to confirm t n rigtppors that the veciors that end near P, are shorter rs that start near p, İhus the net aow is outward near Pi, so div F(P) > 0 Pi is a source. Near Pa, on the other hand, the incoming arrows are longer than the e the net flow is inward,...
Use rectangular, cylindrical and spherical coordinates to set up the triple integrals representing the volume of the region bounded below by the xy plane, bounded above by the sphere with radius and centered at the origin the equation of the sphere is x2 + y2 + z2-R2), and outside the cylinder with the equation (x - 1)2 +y2-1 (5 pts each) Find the volume by solving one of the triple integrals from above.( 5 pts) Total of 20 pts)
Use...
QUESTION 9 A cake is shaped like a hemisphere of radius 4 units with its base on the xy plane (a) Find the volume of the cake using spherical coordinates (5 marks) (b) Now suppose the cake is sliced by a plane perpendicular to the xy -plane at x = a, a > 0 . Let D be the smaller of two pieces produced. Set up a suitable integral for the volume of D (DO NOT EVALUATE). (7 marks)
QUESTION...
1. Use cylindrical coordinates to SET UP the integral for the volume of the portion of the unit ball, 22 +232 + x2 < 1, above the plane z = 12 2. (a) Write in spherical coordinates the equations of the following surfaces: (i) x2 + y2 + x2 = 4 (ii) z = 3x2 + 3y2 (b) SET UP the integral in spherical coordinates for the volume of the solid inside the surface 22 + y2 + x2 =...
3. 3D stuff. cylindrical coordinates. A cone of uniform mass density Po has its tip at the origin and its axis of symmetry is aligned with the z axis. The base of the cone is at H and has radius R. Draw a big picture! Compute the following things a. the total mass of the cone. b. the center of mass of the cone. c. its moment of inertia I2z around the z axis
11. Evaluate S. 'S*(1 + 3x2 + 2y?) dx dy. 12. Find the volume in the first octant of the solid bounded by the cylinder y2 + z2 = 4 and the plane x = 2y. Graph for Problem 12 13. Find the volume under the paraboloid z = 4 - x2 - y2 and above the xy-plane. N Consider the solid region bounded above by the sphere x + y + z = 8 and bounded below by the...
1. Using polar coordinates in the x-y plane, find the volume of the solid above the cone z r and below the hemisphere z= v8-r2. As a check the answer is approximately 13.88 but of course you have to calculate the exact answer 2. At the right is the graph of the 8-leafed rose r 1+2cos(40) Calculate the area of the small leaf. As a check the answer is 0.136 to 3 places of decimal (But of course you have...
Question 2 (1 point) Identify the surface r = 1, in cylindrical coordinates. Plane Cone Half plane Disc Sphere Circle Line segment Cylinder Use spherical coordinates to find the volume of the solid that lies above the cone z = V3x2 + 3y2 and below the sphere x2 + y2 + 2? first octant. Write = 1 in the V = L*S*%' * sin ødpdepdo 1. O 2. 1 d = < 3. À b= 4. 7T 2 5. Ő...
GIVEN: Ω isthe portion of the surface of the sphere centered at the origin of radius 3 above 1.2 1(xy, z) the plane, z-2: Ω: the field F = (x, x,x). a) FIND the flux of VrF through Ω in the given direction: n has positive 2-component. HINT: (radius a)on Q:(spherical coordinates) b) Parameterize the path,c-a2, (r,g,z)asin g dode with orientation to agree with the given n for Ω ANS: (a) 5 c) With positive orientation,an -e DETERMINE: F.ds ANS:...
NO.25 in 16.7 and NO.12 in
16.9 please.
For the vector fied than the vecto and outgoing arrows. Her can use the formula for F to confirm t n rigtppors that the veciors that end near P, are shorter rs that start near p, İhus the net aow is outward near Pi, so div F(P) > 0 Pi is a source. Near Pa, on the other hand, the incoming arrows are longer than the e the net flow is inward,...
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