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Every application of the definite integral

Area, volume, net change, average value — the same integral, read four different ways.

Every application here is the same idea — slice a quantity into infinitesimal pieces, then sum (integrate) them — dressed up for a different geometric or physical question. Area between two curves slices vertically (or horizontally) and integrates the gap. A solid of revolution slices into disks (region touches the axis) or washers (region has a gap from the axis) and integrates cross-sectional area. Net change integrates a rate to recover a total. Average value divides that same integral by the interval's width.

Area

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The region trapped between two graphs, found by integrating the difference of the top and bottom functions.

Example

The area between y=x and y=x^2 on [0,1] is \int_0^1(x-x^2)\,dx=\tfrac16.

Volume of a solid of revolution

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Finding the volume of a solid of revolution by slicing it into circular disks perpendicular to the axis of rotation.

Use when the region touches the axis of revolution directly.

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Finding the volume of a solid of revolution with a hole through it, by slicing into ring-shaped washers.

Use when the region is offset from the axis, leaving a hole.

Net change and average value

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The total change in a quantity over an interval equals the definite integral of its rate of change.

Example

\int_0^5 v(t)\,dt gives net displacement when v is velocity.
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The height of the flat rectangle whose area over an interval matches the actual area under the curve.

Example

The average value of x^2 on [0,3] is \tfrac13\int_0^3 x^2\,dx = 3.

More applications

Graph settings

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