Applications of Integration

Applications of Integration

• Integration is primarily used in calculus to calculate area, volume, and averages.

Area Under a Curve:

• The definite integral of a positive function can give you the area under the curve and above the axis over a given interval.

• To find the area under a curve, integrate the function over a given interval: ∫ from a to b (f(x) dx).

Volume of Revolution:

• Volume of a solid of revolution can be found by using the integration. One of the formulas used is ∫πy²dx, known as the disk method.

• If the solid of revolution has a hole (shell), the formula is 2π∫y h dx, known as the shell method.

Averages:

• To find the mean (average) value of a function over an interval [a, b], use 1/(b-a)∫(from a to b) f(x) dx.

Differential Equations:

• Sometimes, integration is used to solve differential equations. A differential equation is one which is written in terms of derivatives.

• A first order differential equation can generally be written as dy/dx = g(x), and a solution is given by integrating both sides.