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What is Integration?

Integration is one of the two fundamental operations of calculus (the other being differentiation). It's essentially the reverse process of differentiation and is used to find areas, volumes, and accumulated quantities.

Definition of Integration

In calculus, integration is the process of finding the integral of a function. An integral can be thought of as:

  • The reverse of differentiation (finding an antiderivative)
  • The area under a curve
  • The accumulation of quantities over an interval

The integral symbol ∫ was introduced by Leibniz and is an elongated "S" representing "summa" (sum in Latin), reflecting the concept of integration as a sum of infinitesimally small quantities.

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

Where:

  • ∫ is the integral symbol
  • f(x) is the integrand (the function being integrated)
  • dx indicates integration with respect to x
  • F(x) is the antiderivative of f(x)
  • C is the constant of integration

The Relationship Between Derivatives and Integrals

Integration and differentiation are inverse operations. If you differentiate a function and then integrate the result, you get back to the original function (plus a constant).

The Inverse Relationship

$$\frac{d}{dx}\left[\int f(x)\,dx\right] = f(x)$$
$$\int \frac{d}{dx}[F(x)]\,dx = F(x) + C$$

Example:

  • If F(x) = x², then F'(x) = 2x (differentiation)
  • If we integrate 2x, we get x² + C (integration)

Understanding Antiderivatives

An antiderivative of a function f(x) is a function F(x) whose derivative equals f(x).

$$\text{If } F'(x) = f(x), \text{ then } F(x) \text{ is an antiderivative of } f(x)$$

Note that antiderivatives are not unique—if F(x) is an antiderivative of f(x), then so is F(x) + C for any constant C. This is why we always add "+ C" when finding indefinite integrals.

Why the Constant C?

Since the derivative of any constant is zero, when we "reverse" the differentiation process, we lose information about any constant that might have been present. The "+ C" represents all possible constants.

Types of Integrals

There are two main types of integrals:

1. Indefinite Integrals

An indefinite integral has no limits and represents a family of antiderivatives:

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

The result is a function (or family of functions) rather than a number.

2. Definite Integrals

A definite integral has upper and lower limits and represents the net area under a curve:

$$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$

The result is a specific numerical value.

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) establishes the connection between differentiation and integration. It has two parts:

Part 1 (FTC I)

If f is continuous on [a, b], then the function:

$$g(x) = \int_{a}^{x} f(t) \, dt$$

is continuous on [a, b], differentiable on (a, b), and g'(x) = f(x).

Part 2 (FTC II)

If f is continuous on [a, b] and F is any antiderivative of f, then:

$$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$

The FTC is one of the most important theorems in mathematics—it allows us to evaluate definite integrals by finding antiderivatives, rather than computing limits of Riemann sums.

Geometric Interpretation

Geometrically, a definite integral represents the signed area between the curve y = f(x) and the x-axis from x = a to x = b:

  • Area above the x-axis is positive
  • Area below the x-axis is negative

Area Under a Curve

To find the total area (ignoring sign), you may need to split the integral at points where f(x) = 0 and take absolute values of each piece.

Real-World Applications

Integration has countless applications across science, engineering, and economics:

  • Physics: Finding displacement from velocity, work done by a force
  • Engineering: Calculating volumes, centers of mass, moments of inertia
  • Economics: Consumer and producer surplus, present value calculations
  • Statistics: Probability distributions, expected values
  • Biology: Population growth models, drug concentration over time

Next Steps

Now that you understand what integration is, you're ready to learn the specific rules and techniques:

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