The Fundamental Theorem of Calculus: Concepts, Examples, and Applications

5.4 The Fundamental Theorem of Calculus

Introduction to the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus establishes a profound connection between the two main branches of calculus: differential calculus and integral calculus. It precisely describes the inverse relationship between differentiation and integration, showing that these operations essentially undo each other.

Part 1: Functions Defined by Integrals

Consider a function defined by an integral:

• Definition: If f is a continuous function on [a, b], then the function g defined by $g(x) = \int_a^x f(t) \, dt$ depends only on x, which is the variable upper limit of integration.

• Interpretation: If f is positive, g(x) represents the area under the graph of f from a to x. This is often called the "area so far" function.

Example: Calculating Values of g(x)

Let f be the function shown below. Calculate g(0), g(1), g(2), g(3), g(4), and g(5), then sketch a rough graph of g.

• $g(1) = \int_0^1 f(t) \, dt$ is the area of a triangle: $= 1$

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• $g(2) = \int_0^2 f(t) \, dt = g(1) + \int_1^2 f(t) \, dt = 1 + 2 = 3$

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• $g(3) = g(2) + \int_2^3 f(t) \, dt \approx 3 + 1.3 = 4.3$

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• $g(4) = g(3) + \int_3^4 f(t) \, dt \approx 4.3 + (-1.3) = 3.0$

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• $g(5) = g(4) + \int_4^5 f(t) \, dt \approx 3 + (-1.3) = 1.7$

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The graph of g increases while f(t) is positive, attains a maximum at x = 3, and decreases when f(t) becomes negative.

Antiderivatives and the Fundamental Theorem

If f(t) = t and a = 0, then:

• $g(x) = \int_0^x t \, dt = \frac{x^2}{2}$

• The derivative $g'(x) = x$, which is the original function f.

Formal Statement: Fundamental Theorem of Calculus, Part 1

If f is continuous on [a, b], then the function $g(x) = \int_a^x f(t) \, dt$ is continuous and differentiable, and $g'(x) = f(x)$.

Example: Differentiating an Integral Function

Find the derivative of $g(x) = \int_0^x \sqrt{1 + t^2} \, dt$.

• By the Fundamental Theorem, $g'(x) = \sqrt{1 + x^2}$.

Part 2: Evaluation of Definite Integrals

The second part of the Fundamental Theorem provides a practical method for evaluating definite integrals:

• If f is continuous on [a, b], then $\int_a^b f(x) \, dx = F(b) - F(a)$, where F is any antiderivative of f (i.e., $F'(x) = f(x)$).

Differentiation and Integration as Inverse Processes

The two parts of the Fundamental Theorem together show that differentiation and integration are inverse processes:

• If $g(x) = \int_a^x f(t) \, dt$, then $g'(x) = f(x)$.

• If $F'(x) = f(x)$, then $\int_a^b f(x) \, dx = F(b) - F(a)$.

Summary Table: Fundamental Theorem of Calculus

Part	Statement
Part 1	If $g(x) = \int_a^x f(t) \, dt$, then $g'(x) = f(x)$.
Part 2	$\int_a^b f(x) \, dx = F(b) - F(a)$, where $F'(x) = f(x)$.

Key Formulas

• $\frac{d}{dx} \int_a^x f(t) \, dt = f(x)$

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

Conclusion

The Fundamental Theorem of Calculus is central to understanding how integration and differentiation are related. It provides the foundation for evaluating definite integrals and for interpreting integrals as antiderivatives.