Calculus: Integration and Its Applications

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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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