A definite integral as the area under the function between and

The definite integral, when

is the signed area between the function and the x-axis where ranges from to . According to the Fundamental theorem of calculus, if

the definite integral can be calculated by:

To take the definite integral of this function, one would find the antiderivative of the function at and subtract the value of from this. This will be equal to the signed area under the function in .

A definite integral can be thought of as a Riemann sum of infinitely small rectangles, or

If the rectangles are of equal width, then

Integration can be thought of as a generalization of multiplication. If f(x)=constant then the definite integral reduces to ordinary multiplication:


Definite integrals have many applications in geometry, physics, and other fields. For example, volume is the integral of area, and velocity is the integral of acceleration. For instance, using integrals it is possible to calculate the volume of a solid object such as a solid of revolution or a pyramid.

Area under a curve

A definite integral in is equal to the signed area between the curve and the x-axis. For example, to calculate the area under the graph of on the interval , one would first take the integral as follows and evaluate at the end points:

Volume of a solid

See also: Solid of revolution, Volume by cross sections and Multiple integral

Suppose we want to find the volume of a cone with radius and height . We can do this by integrating the circular cross sections from 0 to . The radius will be equal to the distance between the x-axis and the function , so the area of a cross section will be equal to or . Therefore, the volume of the cone will be equal to

Area with polar coordinates

Given the polar function , the area under the function as a Riemann sum is

As an integral, this would be

Arc length

Arc length can be calculated by summing an infinite number of infinitesimally small line segments along a curve. The formula for doing this with regular functions is

With a parametric equation, this is equal to

In polar coordinates, arc length is equal to

See also

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