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 \( f(x) = x^n \), then:
\[ \int x^n dx = \frac{x^{n+1}}{n+1} + C \]
Where \( C \) is the constant of integration.
Constant Rule for Integration
If \( f(x) = c \), where \( c \) is a constant, then:
\[ \int c \, dx = cx + C \]
Sum and Difference Rule
The sum and difference rule for integration states:
\[ \int (f(x) \pm g(x)) \, dx = \int f(x) \, dx \pm \int g(x) \, dx \]
Indefinite Integrals
Example 1:
Find the integral of \( f(x) = 3x^2 \).
\[ \int 3x^2 \, dx = \frac{3x^3}{3} + C = x^3 + C \]
Example 2:
Find the integral of \( f(x) = 2x + 1 \).
\[ \int (2x + 1) \, dx = \int 2x \, dx + \int 1 \, dx = x^2 + x + C \]
Definite Integrals
What Is a Definite Integral?
A definite integral calculates the area under the curve of a function \( f(x) \) over a specific interval \( [a, b] \), and is written as:
\[ \int_a^b f(x) \, dx \]
This represents the total accumulation of the function between the limits \( a \) and \( b \).
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 \( y = x^2 \) between \( x = 0 \) and \( x = 2 \):
\[ \int_0^2 x^2 \, dx = \left[\frac{x^3}{3}\right]_0^2 = \frac{8}{3} – 0 = \frac{8}{3} \]
So, the area under the curve is \( \frac{8}{3} \) square units.
Velocity and Displacement in Science
In Science, integration is used to calculate displacement from velocity. If the velocity \( v(t) \) is given as a function of time, the displacement \( s(t) \) is:
\[ s(t) = \int v(t) \, dt \]
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 \( f(x) = 4x \) between \( x = 1 \) and \( x = 3 \).
Solution:
\[ \int_1^3 4x \, dx = \left[2x^2\right]_1^3 = 18 – 2 = 16 \]
So, the area under the curve is 16 square units.
Common Mistakes in Integration
- Forgetting the Constant of Integration: Always include \( C \) when performing indefinite integrals.
- Incorrect Application of Limits: Be careful to substitute the limits into the antiderivative correctly.
Practice Questions
- Find the integral of \( 5x^3 \) with respect to \( x \).
- Calculate the area under the curve \( y = x^2 \) between \( x = -1 \) and \( x = 1 \).
- A particle’s velocity is given by \( v(t) = 3t^2 \). Find its displacement from \( t = 0 \) to \( t = 4 \).
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