Calculus: Integration and Its Applications
What Is Integration?
Integration is the reverse process of differentiation. While differentiation calculates the rate of change, integration helps to find the total quantity from a rate of change. The result of an integration is often referred to as the antiderivative.
There are two main types of integrals:
- Indefinite Integral: Represents the general form of the antiderivative.
- Definite Integral: Represents the area under a curve within a specific interval.
Basic Integration Rules
Power Rule for Integration
The power rule for integration is the reverse of the power rule for differentiation. If 
, then:
![\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-eac6c067222d5ae64d48d163380ff088_l3.png "Rendered by QuickLaTeX.com")
Where 
is the constant of integration.
Constant Rule for Integration
If 
, where 
is a constant, then:
![\[ \int c \, dx = cx + C \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-7d7ea52b99b7da9d241a99d3dfe1e211_l3.png "Rendered by QuickLaTeX.com")
Sum and Difference Rule
The sum and difference rule for integration states:
![\[ \int (f(x) \pm g(x)) \, dx = \int f(x) \, dx \pm \int g(x) \, dx \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-cbfaa323c925dd74e1ed08519af31149_l3.png "Rendered by QuickLaTeX.com")
Indefinite Integrals
Example 1:
Find the integral of 
.
![\[ \int 3x^2 \, dx = \frac{3x^3}{3} + C = x^3 + C \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-61cc94dbdd470f1ca5ea3b82be77967d_l3.png "Rendered by QuickLaTeX.com")
Example 2:
Find the integral of 
.
![\[ \int (2x + 1) \, dx = \int 2x \, dx + \int 1 \, dx = x^2 + x + C \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-9393bd08c3d19ddab72dc15bd9cd35d0_l3.png "Rendered by QuickLaTeX.com")
Definite Integrals
What Is a Definite Integral?
A definite integral calculates the area under the curve of a function 
over a specific interval ![[a, b]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-1697b646881a88b3a6029defcd0dd8f6_l3.png "Rendered by QuickLaTeX.com")
, and is written as:
![\[ \int_a^b f(x) \, dx \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-a35db2b2d108cbe41ad8d08e4fc263ae_l3.png "Rendered by QuickLaTeX.com")
This represents the total accumulation of the function between the limits 
and 
.
Applications of Integration
Area Under a Curve
Integration is widely used in calculating the area under curves. For example, finding the area under the curve 
between 
and 
:
![\[ \int_0^2 x^2 \, dx = \left[\frac{x^3}{3}\right]_0^2 = \frac{8}{3} - 0 = \frac{8}{3} \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-3f9f38592456bacfd1a0c876dbcdfa32_l3.png "Rendered by QuickLaTeX.com")
So, the area under the curve is 
square units.
Velocity and Displacement in Science
In Science, integration is used to calculate displacement from velocity. If the velocity 
is given as a function of time, the displacement 
is:
![\[ s(t) = \int v(t) \, dt \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-516660c84cf84625bff644e63944e095_l3.png "Rendered by QuickLaTeX.com")
Probability and Statistics
In probability theory, integration is used to find the probability of a continuous random variable falling within a particular range.
Example Problem
Problem: Find the integral of 
between 
and 
.
Solution:
![\[ \int_1^3 4x \, dx = \left[2x^2\right]_1^3 = 18 - 2 = 16 \]](https://skinattuition.co.uk/core/ql-cache/quicklatex.com-909ae94d2c49172f9cd08b82e8fe75fa_l3.png "Rendered by QuickLaTeX.com")
So, the area under the curve is 16 square units.
Common Mistakes in Integration
- Forgetting the Constant of Integration: Always include
when performing indefinite integrals. - Incorrect Application of Limits: Be careful to substitute the limits into the antiderivative correctly.
Practice Questions
- Find the integral of
with respect to 
. - Calculate the area under the curve
between 
and 
. - A particle’s velocity is given by
. Find its displacement from 
to .
with respect to 
.
and 
. Find its displacement from 
to 
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