# A-level Mathematics/CIE/Pure Mathematics 1/Integration

## The Antiderivative

editIntegration is defined as the reverse process of differentiation. Thus, it is the process of finding the **antiderivative** of an expression.

The antiderivative is also called the *integral* of an expression, and is represented using the symbol .

e.g.

### The Constant of Integration

editA problem with integration is that many different expressions have the same derivative, such as and . Expressions with different constant terms may have the same derivative, so when we integrate an expression, we need to add an arbitrary constant to the end, which represents this unknown value.

Therefore,

In some scenarios, we have a point on a curve and an expression for its derivative. From that, we need to find the equation of the curve, which will require us to find the constant of integration by substituting the values from the point.

e.g. The point is on a curve with gradient . Find the equation of the curve.

## Definite Integrals

editA **definite integral** is an integral between two given bounds and . These bounds are written .

For a function with integral , the definite integral

e.g. Find

Note that with definite integrals, the arbitrary constants cancel out. This means we don't actually need to write them when working with definite integrals.

### Improper Integrals

editAn **improper integral** is a definite integral where one of the bounds is invalid.

e.g. is invalid at

To evaluate an improper integral, we need to find the limit of the integral as one of the bounds approaches the value we are looking for.

### Area under a Curve

editA definite integral can be used to find the area under a curve.

e.g. Find the area bounded by , the x-axis, the line and the line

### Solids of Revolution

editA **solid of revolution** is a volume which is obtained by rotating a curve about an axis between two bounds.

The volume can be calculated as the sum of a series of tiny cylinders. If we're rotating about the x-axis, this sum is equal to where is the width of each cylinder. As approaches zero, the sum becomes .