Trivium
Given a solid in with a boundary that can be described analytically either in rectangular, cylindrical, or spherical coordinates, and a function compute the value of
Given an iterated integral of a function on a region and a one-to-one transformation of compute the Jacobian of the transformation and write down the equivalent iterated integral over the pre-transformed domain,
Exercises
- Evaluate the triple integral where is the rectangular prism
- Evaluate where is the solid in the first octant bounded by the surface and the planes and
- Evaluate where is the solid bounded by the paraboloid and the plane
- Sketch the solid over which the iterated integral is concerned, and then rewrite the iterated integral with the order of the variables permuted: first with respect to then with respect to
- Calculate the volume of the tetrahedron bound by the planes and and and
- Calculate the coordinates of the center of mass of the solid with constant density that is bounded by the parabolic cylinder and the planes and and
- Evaluate for the solid that lies under the paraboloid and above the -plane.
- A solid lies within the cylinder on the positive side of the -plane, below the plane and above the paraboloid The density at any point in the solid is three times its distance to the -axis. What is the mass of
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Evaluate this iterated integral:
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For the unit ball
evaluate
- Calculate the volume of the solid that lies above the cone but below the sphere
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Consider the following change of coordinates
and calculate the corresponding Jacobian.
Problems & Challenges
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Consider the tetrahedron in with vertices
and and and .
- Write down three integrals — one each based on rectangular, cylindrical, and spherical coordinates — that express the volume of the tetrahedron. Then evaluate one of them to calculate the volume.
- What are the coordinates of the centroid of this tetrahedron?
- What is the surface area of this tetrahedron?
- There exists a change of variables, a transformation of space, that maps the vertices of this tetrahedron to the points and and and . What is this transformation, and what is its Jacobian? Use the transformation and Jacobian to compute the volume of the original tetrahedron.
- James Stewart Show that So far, no one has been able to express the value of this integral/sum exactly in terms of already-known constants.