Integration is a fundamental concept in calculus that involves finding the area under a curve.
It is the opposite of differentiation, which involves finding the slope or rate of change of a
function. Integration can be used to solve a variety of problems, from finding the
displacement of a moving object to calculating the volume of a solid.
The process of integration can be broken down into several steps:
Identify the function that needs to be integrated. This is typically represented by the
function f(x) or y.
Determine the limits of integration. This is represented by the lower limit, a, and the upper
limit, b. The area under the curve between these two limits is what we are trying to find.
Choose an appropriate method of integration. There are several methods of integration,
including:
Antiderivative: This is the most common method of integration. It involves finding the
antiderivative of the function, which is the inverse of differentiation. The antiderivative of a
function is represented by the symbol ∫.
Substitution: This method involves making a substitution in the original function to simplify
it. This is often used when the function is in a form that is difficult to integrate directly.
Integration by parts: This method involves breaking the function into two parts and then
integrating each part separately. It is used when the function is a product of two functions,
one of which is easy to integrate and the other is not.
Trigonometric integration: This method is used to integrate trigonometric functions such as
sine, cosine, and tangent.
Evaluate the integral. This involves calculating the definite integral, which is the difference
between the antiderivative at the upper limit and the antiderivative at the lower limit. The
definite integral is represented by the symbol ∫b a f(x)dx.
Check the result. This involves ensuring that the result is in the correct form, has the correct
units, and makes sense in the context of the problem.
