Fundamental theorem of calculus

The fundamental theorem of calculus is a theorem that connects two big ideas in math: differentiating a function and integrating a function. Differentiating tells us how a function changes at each point, like finding the slope of a line. Integrating helps us find the total area under a curve.

The theorem shows that these two processes are opposites of each other.

The first part of the theorem says that for a continuous function f, we can find something called an antiderivative by integrating the function over an interval. This means we can use integration to rebuild the original function from its rate of change.

The second part of the theorem tells us that to find the total area under a function f over a fixed interval, we just need to see how an antiderivative F changes between the two ends of that interval. This makes calculating areas much easier if we can find the antiderivative using symbolic integration, instead of using harder numerical integration methods.

History

Sketch of geometric proof

The first fundamental theorem of calculus shows how two important ideas in math are connected: finding slopes and finding areas under curves.

Imagine you have a curve on a graph. You can create an "area function" that tells you the area under the curve up to any point. By looking at small parts of the area, we find that the slope of this area function at any point is the same as the value of the original curve at that point. This means that taking the derivative of an area (an integral) gives you back the original function. In other words, derivatives and integrals are like opposite operations, undoing each other. This is the core idea of the Fundamental Theorem of calculus.

Intuitive understanding

The fundamental theorem of calculus shows that integration and differentiation are opposite actions. Imagine you know how fast a car is moving at each moment. By adding up all the small distances it moves each second, you can find the total distance it has traveled. This adding up is called integration.

The theorem also says that if you know the total distance traveled up to any point, the rate at which the distance changes at that point is the speed of the car. Integrating gives the total distance, and differentiating shows the speed.

Formal statements

The Fundamental Theorem of Calculus has two main parts that connect two big ideas in math: derivatives and integrals.

Fundamental theorem of calculus (animation)

The first part says that if you have a special kind of function, you can find its antiderivative by integrating it. This means that integrating and then taking the derivative brings you back to your original function.

The second part shows that if you know an antiderivative of a function, you can easily find the value of a definite integral by using the antiderivative at the endpoints of the interval. This part is useful even if the function isn’t perfectly continuous everywhere.

antiderivative definite integrals

Proof of the first part

To understand the first part of the Fundamental Theorem of Calculus, we look at a special function ( F ) defined by an integral. For any two points ( x_1 ) and ( x_1 + \Delta x ), the difference between ( F ) at these points can be shown using the integral of the function ( f ) between them.

By the mean value theorem for integration, there is a point ( c ) between ( x_1 ) and ( x_1 + \Delta x ) where this integral equals ( f(c) ) times ( \Delta x ). As ( \Delta x ) gets very small, the ratio of the change in ( F ) to ( \Delta x ) approaches ( f(c) ). Because ( c ) stays close to ( x_1 ) as ( \Delta x ) shrinks, this ratio ultimately equals ( f(x_1) ). This shows that the derivative of ( F ) at ( x_1 ) is ( f(x_1) ). This connects integration and differentiation, showing they are opposite operations.

mean value theorem for integration squeeze theorem

Proof of the corollary

The proof shows that if we have a function f that stays smooth between two points, a and b, we can find another function F to help us find the total area under f between those points. We do this by comparing two ways to describe the same idea. One way uses a special function G that adds up small pieces of f. The other way uses F, which we already know works well. By looking closely, we see they are almost the same, just shifted by a fixed amount. This gives us a useful result: the total area under f from a to b is simply the difference between the values of F at those two points.

Main article: mean value theorem
Main article: constant function

Proof of the second part

A converging sequence of Riemann sums. The number in the upper left is the total area of the blue rectangles. They converge to the definite integral of the function.

This proof uses a method called Riemann sums. It connects two big ideas in math: taking the slope of a line (called differentiation) and finding the area under a curve (called integration).

The proof starts with a special math rule called the mean value theorem. This rule helps us break down a complicated problem into smaller, easier pieces. By adding up these small pieces and making them closer together, we get a very accurate answer. In the end, this shows that the difference between two values of a function is the same as the integral — the total area under the curve — between those two points. This links differentiation and integration together in a clear way.

F ( b ) − F ( a ) = ∑ i = 1 n [ F ( x i ) − F ( x i − 1 ) ] . {\displaystyle F(b)-F(a)=\sum _{i=1}^{n}[F(x_{i})-F(x_{i-1})].}

F ( b ) − F ( a ) = ∑ i = 1 n [ f ( c i ) ( Δ x i ) ] . {\displaystyle F(b)-F(a)=\sum _{i=1}^{n}[f(c_{i})(\Delta x_{i})].}

Relationship between the parts

The two parts of the Fundamental Theorem of Calculus are closely connected. If we know one part, we can often understand the other better. For example, if we have a function f and we know one of its antiderivatives G, then we can describe the integral of f from a point a to another point x using G. This shows that integrating and finding antiderivatives are almost opposite operations.

However, it’s important to remember that not every function has a simple antiderivative we can write down. Some functions can be integrated even though we can’t find a simple expression for their antiderivatives. Also, some functions might have antiderivatives but still not be integrable in the usual sense. This shows how the two parts of the theorem fit together in a careful way.

Examples

The Fundamental Theorem of Calculus shows how two big ideas in math—finding slopes of lines and finding areas under curves—are connected. It tells us these two processes are opposites of each other.

One example shows how to find the area under a curve between two points. By using a special formula, we can find the exact area without hard calculations. Another example shows how the theorem helps us understand how functions change over time, linking accumulation to instantaneous change.

Variations in terminology

Different math books use different names for the parts of the fundamental theorem of calculus. Some call the first part the "first fundamental theorem." This part says that if you have a continuous function, you can find a new function by integrating it over an interval.

Other books might call this the "fundamental theorem" and use a different name for the second part. Old math history books also change how they name these ideas, so it can get a little confusing. But no matter what it's called, the ideas connect slopes and areas in smart ways.

Generalizations

The Fundamental Theorem of Calculus can work with more than just smooth functions. Even if a function changes a lot, we can still learn important things. For example, if a function is smooth at one spot in a range, we can make a new function by adding up its values, and this new function will change smoothly at that spot.

The theorem also helps with more complicated functions and calculations. In higher dimensions, ideas from the theorem help us understand how averages of functions behave, and they connect to important results like the divergence theorem and gradient theorem. These ideas show how the Fundamental Theorem of Calculus is important in many parts of mathematics.

Main article: Lebesgue's differentiation theorem

Main articles: Divergence theorem, Gradient theorem

Further information: Generalized Stokes theorem