Download Activity: Volume of Known Cross Sections Calculus allows us to find

Activity: Volume of Known Cross Sections
Calculus allows us to find the volume of solids that are not “normal” shapes. It has two techniques: volume by crosssections and volume by rotation.
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What do various solids look like when you define them by their cross-sections?
How do you write the volume formulas in terms of a function?
How can you estimate the volume? How do you find the exact volume?
PART 1: Simple Prisms
Use your play-doh to create a solid on the above picture that has these characteristics…
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The base of the solid is bounded by both the x- and y-axis, the line x = 10, and the function f(x) = 5.
Every cross-section in the solid is a square that is perpendicular to the x-axis.
You should end up with a rectangular prism.
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Using geometry, how would you find the volume of the prism? (formula only, please)
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Now re-write your formula in terms of x and f(x).
Same picture, new solid…
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The base of the solid is bounded by both the x- and y-axis, the line x = 10, and the function f(x) = 5.
Every cross-section in the solid is an equilateral triangle that is perpendicular to the x-axis.
You should end up with a triangular prism.
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Using geometry, how would you find the volume of the prism? (formula only, please) (HINT: area of an
equilateral triangle =
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√
)
Now re-write your formula in terms of x and f(x).
PART 2: Not Prisms Anymore!
Use your play-doh to create a solid on the above picture that has these characteristics…
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The base of the solid is bounded by the x-axis and the function f(x) = -x2 + 5x.
Every cross-section in the solid is an equilateral triangle that is perpendicular to the x-axis.
You should not end up with a prism… in fact, it shouldn’t be anything you’ve seen in geometry!
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Make a few careful slices in your solid by keeping your knife perpendicular to the x-axis.
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Look at the face (the side the knife was on) – you should see an equilateral triangle on every slice… but not the
same size triangle each time! (if not, remake your solid and try again)
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Smoosh it back together and make equal slices along the grid lines. Separate the pieces – how could you
estimate the volume of one piece? (formula only, please) (HINT: Use a middle piece.)
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Now re-write that formula in terms of x and f(x)
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How could you estimate the volume of the entire solid by using those pieces?
Repeat this part with the following cross-sections…
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Semi-circles
Squares
45-45-90 triangles with a leg on the base
45-45-90 triangles with the hypotenuse on the base
Answer the same questions in your notes.
***What do ALL of these volume formulas have in common?***
PART 3: Where’s the calculus?
I want to find the volume of a solid whose base is bounded by the x-axis, the line x = 6, and the function shown above.
The cross-sections of this solid are squares perpendicular to the x-axis.
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Model the scenario with the play-doh.
Estimate the volume like you did in part 2.
o First write a formula for the volume of one piece in terms of x and f(x).
o Then write how you’d estimate the total volume.
o Now actually estimate the volume.
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Volume estimate = _____________________________
How can you make your estimation more accurate?
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How can you make it a lot more accurate?
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What calculus idea is this implying? Explain.
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Write a calculus expression using all of this information that would find the volume, then find the volume.
Repeat this part with the following cross-sections…
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Semi-circles, Squares, 45-45-90 triangles with a leg on the base
Answer the same questions in your notes
***How do you find the volume no matter what shape the cross-sections are?***
PART 4: The problems
These are past AP problems. You may use the play-doh if you’d like.