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Volumes of a Solid

Volumes of a Solid. AP Topic. Applications of integrals Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should

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Volumes of a Solid

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  1. Volumes of a Solid

  2. AP Topic Applications of integrals Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems. Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region, the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, and accumulated change from a rate of change.

  3. Definition: Volume of a Solid • The volume of a solid of known integrable cross section are A(x) from x = a to x = b is the integral of A from a to b.

  4. How to find Volume by slicing 1) Sketch the solid and a typical cross section. 2) Find a formula for A(x). 3) Find the limits of integration. 4) Integrate A(x) to find the volume.

  5. A pyramid 3m high has congruent triangular sides and a square base that is 3 m on each side. Each cross section of the pyramid parallel to the base is a square. Find the volume.

  6. Solid of revolution http://clem.mscd.edu/~talmanl/MathAnim.html

  7. Circular Cross Sections • Find the volume of the solid generated by revolving the region bounded by

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