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AP Calculus AB: 10.5.1 Finding Volumes Using Cross-Sectional Slices

Mathematics8 CardsCreated 3 months ago

This set covers how to find the volume of solids by integrating the area of cross-sectional slices. It explains the concept of slicing objects into thin sections, setting up integrals based on cross-sectional area functions, and applying these principles to solids with vertical or horizontal slices.

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Finding Volumes Using Cross-Sectional Slices

  • The volume of a solid with vertical cross-sections of area A(x) is V, where

  • The volume of a solid with horizontal cross-sections of area A(y) is V, where

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Key Terms

Term
Definition

Finding Volumes Using Cross-Sectional Slices

  • The volume of a solid with vertical cross-sections of area A(x) is V, where

  • The volume of a solid with horizontal cross-sect...

note

  • Finding the volume of an object is pretty easy if you know the formula. But a lot of objects don’t fit any commonly known formulas. How do ...

Suppose you are told that a particular solid is h feet tall and that the cross-sections of the solid perpendicular to the height are circles of diameter equal to the height at each cross section. Which of the following formulas would correctly compute the volume of this figure?

V=πh^3/12

Which of the following ways of slicing this object would not be appropriate for finding its volume?

To find volume, slicing should be perpendicular to the axis of the solid.

Inappropriate slicing:

Which of the following produces the volume of a solid that lies alongside the interval [a, b] on the x‑axis and has the continuous cross-sectional area function A (x)?

V=∫^b _a A(x)dx

Which of the following statements accurately compares the volumes of the following two solids, which have the same height and the same horizontal cross-sectional slices?

VA = VB

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AP Calculus AB: 10.5.1 Finding Volumes Using Cross-Sectional Slices
TermDefinition

Finding Volumes Using Cross-Sectional Slices

  • The volume of a solid with vertical cross-sections of area A(x) is V, where

  • The volume of a solid with horizontal cross-sections of area A(y) is V, where

note

  • Finding the volume of an object is pretty easy if you know the formula. But a lot of objects don’t fit any commonly known formulas. How do you find the volumes of these strangely shaped objects?

  • If you consider the object in slices, then all you would have to do is find the volume of each individual slice and then add them together. These slices are called cross-sections.

  • This process is actually a pretty good way to calculate the volume of an object when the cross-sections are easily defined. Consider a hunk of cheese. If you slice it up, the individual slices look like rectangular prisms.

  • Finding volumes is one of the applications of the definite integral.

  • Suppose you are given an object and you can define the cross sections by a function in terms of x.

  • The volume of the object on the interval [a, b] is given by the definite integral to the left. All the integral does is sum up an infinite number of slices of the volume, each of a very tiny width called dx.

  • The same process can be applied to an object whose cross sections are defined by a function in terms of y.

Suppose you are told that a particular solid is h feet tall and that the cross-sections of the solid perpendicular to the height are circles of diameter equal to the height at each cross section. Which of the following formulas would correctly compute the volume of this figure?

V=πh^3/12

Which of the following ways of slicing this object would not be appropriate for finding its volume?

To find volume, slicing should be perpendicular to the axis of the solid.

Inappropriate slicing:

  • Slicing parallel to the axis (won’t give cross-sectional areas to integrate)

  • Irregular or non-uniform slices that don’t represent consistent cross-sections

Which of the following produces the volume of a solid that lies alongside the interval [a, b] on the x‑axis and has the continuous cross-sectional area function A (x)?

V=∫^b _a A(x)dx

Which of the following statements accurately compares the volumes of the following two solids, which have the same height and the same horizontal cross-sectional slices?

VA = VB

Which of the following is the volume of a cylinder with radius r and height h ?

V = π r 2h

Which of the following produces the volume of a solid that lies alongside the interval [c, d ] on the y‑axis and has the continuous cross-sectional area function A ( y)?

V=∫^d _c A(y)dy