Integral: A Detailed Explanation

An integral is a fundamental concept in calculus that represents the accumulation of quantities, such as areas under curves, total accumulated change, or volumes. Integrals can be broadly classified into two types:

1. Definite Integral
2. Indefinite Integral

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1. Indefinite Integral

An indefinite integral represents the family of functions whose derivative is the given function. It is written as:

$\int f(x) \, dx = F(x) + C$

where:

• $f(x)$ is the function to be integrated, called the integrand.
• $F(x)$ is the antiderivative of $f(x)$.
• $C$ is the constant of integration, since integration is the inverse of differentiation and the derivative of a constant is zero.

Example: $\int x^2 \, dx = \frac{x^3}{3} + C$

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2. Definite Integral

A definite integral computes the net area under a curve between two points $a$ and $b$. It is written as:

$\int_a^b f(x) \, dx$

This represents the limit of a Riemann sum, meaning the sum of infinitely small areas. The Fundamental Theorem of Calculus states:

$\int_a^b f(x) \, dx = F(b) - F(a)$

where $F(x)$ is the antiderivative of $f(x)$.

Example: $\int_1^3 x^2 \, dx$

Step 1: Compute the antiderivative of $f(x) = x^2$: $F(x) = \frac{x^3}{3}$

Step 2: Evaluate at limits $a = 1$ and $b = 3$:

$F(3) - F(1) = \left( \frac{3^3}{3} \right) - \left( \frac{1^3}{3} \right) = \left( \frac{27}{3} - \frac{1}{3} \right) = \frac{26}{3}$

Thus,
$\int_1^3 x^2 \, dx = \frac{26}{3}$

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Properties of Integrals

1. Linearity:
   $\int \left( a f(x) + b g(x) \right) dx = a \int f(x) dx + b \int g(x) dx$
2. Reversal of Limits (Definite Integral):
   $\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx$
3. Additivity:
   $\int_a^b f(x) dx + \int_b^c f(x) dx = \int_a^c f(x) dx$
4. Integral of a Constant:
   $\int a \, dx = ax + C$

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Applications of Integrals

1. Finding Area under curves.
2. Computing Volumes using methods like disk/washer and shell methods.
3. Solving Differential Equations in physics and engineering.
4. Calculating Work and Energy in physics.
5. Probability and Statistics for cumulative distribution functions.

Would you like a specific example or more details?

Related Questions

1. How do you solve integrals using substitution?
2. What is integration by parts, and how is it used?
3. How do definite integrals relate to the Fundamental Theorem of Calculus?
4. How do improper integrals work, and when do they converge?
5. What are some real-world applications of integrals?

Tip:

When solving integrals, always check if substitution or by-parts methods can simplify the problem before solving directly!