Calculus: Volumes of Solids with Known Cross Sections - An Exploration

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1.39 MB   |   19 pages

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This lesson has recently been revised and redesigned. Students make 8 different shapes and work through a 4 part activity sheet to develop the concept of finding volume of a solid if you know the shape of the cross section.

Students have difficulty picturing solids that are created on a region. This activity has students build 8 different solids on the same base. As students work together to complete all 8 shapes and answer a series of questions they find that it isn't hard to find the volume of a solid. When moving from one shape to another, the students find the only difference is the expression used for the area of the cross section. Students will start with a region bounded by the square root of x, the x-axis and the line x = 4. Then they build various solids by placing cross sections at various x values. All of a sudden the solid takes on a visual. Students work together to find their eight volumes. They write formulas for area, write Riemann sums for the volume, estimate the volume, write an integral for the volume, and calculate the volume.

Students can compare their volumes when they are finished to see which truly holds the smallest to largest volumes. They will also look for relationships between their volumes.

The patterns come in unshaded shapes so you can use colored card stock to duplicate shapes.

A complete set of answers and solutions are include.

Students will find out that finding volume is not so difficult.

After you use it you will want to purchase a copy for a colleague.
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19
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Calculus:  Volumes of Solids with Known Cross Sections - A
Calculus:  Volumes of Solids with Known Cross Sections - A
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