AP®MathsCollege BoardCalculus ABRevision NotesUnit 8: Applications of IntegrationUnit 8 OverviewUnit 8 Summary

Unit 8 Summary (College Board AP® Calculus AB): Revision Note

Written by: Roger B

Reviewed by: Jamie Wood

Updated on 18 May 2026

Applications of integration summary

Key definitions

The average value of a continuous function $f$ over the interval $[a, b]$ is $\frac{1}{b - a} \int_{a}^{b} f (x) d x$
The net change of $f$ between $x = x_{1}$ and $x = x_{2}$ is given by $\int_{x_{1}}^{x_{2}} f' (x) d x$

Key formulas

Let $x (t)$ , $v (t)$ and $a (t)$ represent the position, velocity, and acceleration of an object at time $t$
- $v (t) = \int a (t) d t$ and $v (t) = v (t_{0}) + \int_{t_{0}}^{t} a (w) d w$
- $x (t) = \int v (t) d t$ and $x (t) = x (t_{0}) + \int_{t_{0}}^{t} v (w) d w$
The change in velocity between $t = t_{1}$ and $t = t_{2}$ is $\int_{t_{1}}^{t_{2}} a (t) d t$
The change in position (displacement) between $t = t_{1}$ and $t = t_{2}$ is $\int_{t_{1}}^{t_{2}} v (t) d t$
The distance traveled between $t = t_{1}$ and $t = t_{2}$ is $\int_{t_{1}}^{t_{2}} |v (t)| d t$
The table below shows the key formulas for areas

Area	Formula	Notes
Between: $y = f (x)$ the $x$ -axis the vertical lines $x = a$ and $x = b$	$\int_{a}^{b} \|f (x)\| d x$	Absolute value is not needed if $f (x) \geq 0$ over the interval $[a, b]$
Between: $x = g (y)$ the $y$ -axis the horizontal lines $y = a$ and $y = b$	$\int_{a}^{b} \|g (y)\| d y$	Absolute value is not needed if $g (y) \geq 0$ over the interval $[a, b]$
Between: $y = f (x)$ $y = g (x)$ the vertical lines $x = a$ and $x = b$	$\int_{a}^{b} \|f (x) - g (x)\| d x$	Absolute value is not needed if $f (x) \geq g (x)$ over the interval $[a, b]$
Between: $x = f (y)$ $x = g (y)$ the horizontal lines $y = a$ and $y = b$	$\int_{a}^{b} \|f (y) - g (y)\| d y$	Absolute value is not needed if $f (y) \geq g (y)$ over the interval $[a, b]$

The table below shows the key formulas for volumes using the disc method

Volume	Formula
Around the $x$ -axis Between: $y = f (x)$ the $x$ -axis the vertical lines $x = a$ and $x = b$	$π \int_{a}^{b} {(f (x))}^{2} d x$
Around the $y$ -axis Between: $x = g (y)$ the $y$ -axis the horizontal lines $y = a$ and $y = b$	$π \int_{a}^{b} {(g (y))}^{2} d y$
Around the line $y = k$ Between: $y = f (x)$ the horizontal line $y = k$ the vertical lines $x = a$ and $x = b$	$π \int_{a}^{b} {(f (x) - k)}^{2} d x$
Around the line $x = k$ Between: $x = g (y)$ the vertical line $x = k$ the horizontal lines $y = a$ and $y = b$	$π \int_{a}^{b} {(g (y) - k)}^{2} d y$

The volume of a solid with a cross-section area given by the continuous function $A (x)$ over $[a, b]$ is $\int_{a}^{b} A (x) d x$
The table below shows the key formulas for volumes using the washer method

Volume	Formula	Notes
Around the $x$ -axis Between: $y = f (x)$ $y = g (x)$ the vertical lines $x = a$ and $x = b$	$π \int_{a}^{b} \|{(f (x))}^{2} - {(g (x))}^{2}\| d x$	Absolute value is not needed if $\|f (x)\| \geq \|g (x)\|$ over the interval $[a, b]$
Around the $y$ -axis Between: $x = f (y)$ $x = g (y)$ the horizontal lines $y = a$ and $y = b$	$π \int_{a}^{b} \|{(f (y))}^{2} - {(g (y))}^{2}\| d y$	Absolute value is not needed if $\|f (y)\| \geq \|g (y)\|$ over the interval $[a, b]$
Around the line $y = k$ Between: $y = f (x)$ $y = g (x)$ the vertical lines $x = a$ and $x = b$	$π \int_{a}^{b} \|{(f (x) - k)}^{2} - {(g (x) - k)}^{2}\| d x$	Absolute value is not needed if $\|f (x) - k\| \geq \|g (x) - k\|$ over the interval $[a, b]$
Around the line $x = k$ Between: $x = f (y)$ $x = g (y)$ the horizontal lines $y = a$ and $y = b$	$π \int_{a}^{b} \|{(f (y) - k)}^{2} - {(g (y) - k)}^{2}\| d y$	Absolute value is not needed if $\|f (y) - k\| \geq \|g (y) - k\|$ over the interval $[a, b]$

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Volume	Formula	Notes
Around the $x$ -axis Between: $y = f (x)$ $y = g (x)$ the vertical lines $x = a$ and $x = b$	$π \int_{a}^{b} \|{(f (x))}^{2} - {(g (x))}^{2}\| d x$	Absolute value is not needed if $\|f (x)\| \geq \|g (x)\|$ over the interval $[a, b]$
Around the $y$ -axis Between: $x = f (y)$ $x = g (y)$ the horizontal lines $y = a$ and $y = b$	$π \int_{a}^{b} \|{(f (y))}^{2} - {(g (y))}^{2}\| d y$	Absolute value is not needed if $\|f (y)\| \geq \|g (y)\|$ over the interval $[a, b]$
Around the line $y = k$ Between: $y = f (x)$ $y = g (x)$ the vertical lines $x = a$ and $x = b$	$π \int_{a}^{b} \|{(f (x) - k)}^{2} - {(g (x) - k)}^{2}\| d x$	Absolute value is not needed if $\|f (x) - k\| \geq \|g (x) - k\|$ over the interval $[a, b]$
Around the line $x = k$ Between: $x = f (y)$ $x = g (y)$ the horizontal lines $y = a$ and $y = b$	$π \int_{a}^{b} \|{(f (y) - k)}^{2} - {(g (y) - k)}^{2}\| d y$	Absolute value is not needed if $\|f (y) - k\| \geq \|g (y) - k\|$ over the interval $[a, b]$