Introduction to Integral Calculus

Learning Objectives

Define the indefinite integral as the antiderivative of a function.

Calculate basic indefinite integrals using the reverse Power Rule.

Define the definite integral as the net area under a curve.

State the Fundamental Theorem of Calculus, which connects differentiation and integration.

The Other Side of Calculus: Integration

While differential calculus is about finding rates of change (slopes), integral calculus is about accumulating quantities, most often visualized as finding the area under a curve.

The Indefinite Integral: The Antiderivative

The indefinite integral of a function f(x), denoted ∫f(x)dx, asks the question: 'What function, when differentiated, gives me f(x)?' For this reason, it is also called the antiderivative.

The Reverse Power Rule:

Just as we have a power rule for derivatives, we have one for integrals.

To find the integral of xⁿ:

∫xⁿ dx = (xⁿ⁺¹) / (n+1) + C

In words: add one to the exponent, then divide by the new exponent.

The Constant of Integration, C:

Notice the '+ C'. When we differentiate a constant, it becomes zero. This means there are infinitely many functions that have the same derivative (e.g., the derivatives of x² + 2, x² - 10, and x² are all 2x). The '+ C' represents this unknown constant and must always be included in an indefinite integral.

Example: Find ∫4x³ dx.

Using the reverse power rule: 4 (x³⁺¹) / (3+1) = 4 (x⁴ / 4) = x⁴.

Add the constant: ∫4x³ dx = x⁴ + C.

Check by differentiating: The derivative of x⁴ + C is indeed 4x³.

The Definite Integral: Area Under a Curve

The definite integral, denoted with limits of integration, calculates the net area between the function's graph and the x-axis over a specific interval [a, b].

∫ₐᵇ f(x)dx

This value is a number, not a function. Areas above the x-axis are positive, and areas below are negative.

The Fundamental Theorem of Calculus

This is one of the most important theorems in all of mathematics because it provides the bridge connecting differential and integral calculus. It states that we can evaluate a definite integral by using the antiderivative.

If F(x) is an antiderivative of f(x), then:

∫ₐᵇ f(x)dx = F(b) - F(a)

In words: to find the area under f(x) from a to b, find the antiderivative F(x), evaluate it at b and a, and subtract the results. This is a much easier method than the original technique of summing up an infinite number of tiny rectangles (Riemann sums).

Example: Find the area under the curve f(x) = 2x from x=1 to x=3.

1.Find the antiderivative, F(x): ∫2x dx = x². (We can ignore the '+C' for definite integrals).

2.Evaluate F(b) - F(a): F(3) - F(1) = (3)² - (1)² = 9 - 1 = 8.

3.The area is 8.

Key Terms

Indefinite Integral: The antiderivative of a function, representing a family of functions whose derivatives are the original function. It includes a constant of integration, C.
Definite Integral: A mathematical concept that represents the net area under the curve of a function over a specified interval [a, b]. It evaluates to a single numerical value.
Fundamental Theorem of Calculus: The theorem that links the concepts of differentiating a function (finding its slope) and integrating a function (finding the area under its curve).
Antiderivative: A function F whose derivative is equal to a given function f.