Volumes from Areas of Known Cross Sections (College Board AP® Calculus AB)

Study Guide

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Written by: Roger B

Reviewed by: Dan Finlay

Volumes from areas of known cross sections

How do I find the volume of a solid with known cross sections?

If the area of the cross section of a solid
- can be expressed as a function of $x$
  - i.e. as a function $A$ that may be written in the form $A (x)$
- and if $A (x)$ is continuous on the interval $[a, b]$
Then the volume of the corresponding solid from $x = a$ to $x = b$ is
- $Volume = \int_{a}^{b} A (x) d x$
For example the cross-sectional area of a solid is given by $A (x) = \frac{1}{x + 1}, 0 \leq x \leq 3$
- The volume of the solid between $x = 0$ and $x = 3$ is
  - $Volume = \int_{0}^{3} \frac{1}{x + 1} d x = {[\ln (x + 1)]}_{0}^{3} = \ln 4 - \ln 1 = \ln 4 - 0 = \ln 4$
This method of finding volumes uses the idea of a definite integral as calculating an accumulation of change
- $A (x) \cdot ∆ x$ is the volume of a solid with cross-sectional area $A (x)$ and length $∆ x$
- $A (x) d x$ is the limit of this volume element as $∆ x \to 0$
- The integral $\int_{a}^{b} A (x) d x$ sums up all these infinitesimal volume elements between $x = a$ and $x = b$

Worked Example

The area, in square feet, of the the horizontal cross section of a water tank at height $h$ feet is modeled by the function $f$ given by $f (h) = \frac{50}{e^{h}}$ . The tank has a height of 10 feet.

Based on this model, find the volume of the tank. Indicate units of measure.

Answer:

Use $Volume = \int_{a}^{b} A (x) d x$

$\begin{array}{rcl} Volume & = & \int_{0}^{10} \frac{50}{e^{h}} d h \\ = & 50 \int_{0}^{10} e^{- h} d h \\ = & 50 {[- e^{- h}]}_{0}^{10} \\ = & 50 (- e^{- (10)} - (- e^{- (0)})) \\ = & 50 (1 - e^{- 10}) \\ = & 49.9977 . . . \end{array}$

Don't forget to give the units in the final answer

Area has units of feet squared, and height has area of feet, so the units of volume are feet cubed (or cubic feet)

49.998 cubic feet (to 3 decimal places)

Last updated: 25 September 2024

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