The Fundamental Theorem of Calculus links two big ideas in calculus: accumulation and rate of change. Definite integrals measure accumulated quantity such as area, distance, or total change, while derivatives measure instantaneous change or slope. This theorem matters because it shows these two processes are inverse operations.
It turns many hard area problems into easier function evaluations.
The theorem has two main parts. Part 1 says that if , then , so the derivative of an accumulation function gives back the original function. Part 2 says that if is any antiderivative of , then .
Together, these results explain why shaded area, changing upper limits, and tangent slopes can all be represented in one connected picture.
Understanding Fundamental Theorem of Calculus
An accumulation function records what has built up from one fixed starting point to a moving endpoint. Imagine sliding the endpoint a tiny distance to the right. The new total differs from the old total by one very thin strip under the graph.
Its height is close to the function value at that endpoint. Its width is the tiny movement. The change in accumulated amount divided by the tiny movement is therefore close to the function value.
As the movement becomes extremely small, the approximation becomes exact. This is the central reason the first part works. A continuous graph has no jumps that would disrupt this local strip idea.
The second part depends on a useful cancellation process. An antiderivative describes total change through its output values. When only the change from a starting input to an ending input matters, every contribution between them is included once in the running total.
Evaluating the antiderivative at the ending input captures the whole build up from a reference point. Evaluating it at the starting input captures the part before the interval begins. Subtracting removes that earlier part.
Any constant added to an antiderivative disappears during this subtraction. This is why students may choose any correct antiderivative when evaluating a definite integral.
Units give the theorem a practical meaning. If velocity is measured in meters per second, integrating velocity over seconds gives meters. The result is displacement, meaning final position minus initial position.
A car moving backward has negative velocity, so it reduces displacement. If the goal is total distance traveled, split the interval wherever velocity changes sign, then add the sizes of the separate displacements.
Similar reasoning appears with water flow measured in liters per minute, electric current measured in coulombs per second, and profit rate measured in dollars per day. The integral gives the net accumulated change, not automatically the total amount of activity.
Careful notation prevents many errors. The lower and upper bounds tell where accumulation starts and stops. Reversing their order reverses the sign of the answer.
A graph below the horizontal axis contributes negatively, even if the geometric region has positive physical area. When finding an antiderivative, differentiate it afterward to check that it returns the original function. When the upper endpoint itself depends on the input, the rate of accumulation needs an extra adjustment from the chain rule.
The function value is taken at the moving endpoint, then multiplied by how fast that endpoint moves. Finally, keep the meaning of the answer in view. A numerical result without units or a clear interpretation can hide a correct calculation used for the wrong quantity.
Key Facts
- FTC Part 1: If , then .
- FTC Part 2: If , then .
- An antiderivative of is any function such that .
- The variable inside the integral is a dummy variable, so the name does not affect the value.
- A definite integral gives signed area, so regions below the -axis contribute negative value.
- Average value of on is times .
Vocabulary
- Definite integral
- A definite integral gives the net accumulated value of a function over an interval.
- Antiderivative
- An antiderivative of f is a function whose derivative is f.
- Accumulation function
- An accumulation function is formed by integrating from a fixed starting point to a variable endpoint.
- Dummy variable
- A dummy variable is the integration variable inside an and its name does not affect the value.
- Signed area
- Signed area counts area above the x-axis as positive and area below the x-axis as negative.
Common Mistakes to Avoid
- Confusing the upper limit x with the integration variable, which is wrong because the inside variable is just a placeholder and should be different from the outside variable.
- Forgetting that the definite integral gives signed area, which is wrong because parts of the graph below the -axis subtract from the total instead of adding.
- Using instead of , which is wrong because the Fundamental Theorem uses upper endpoint minus lower endpoint in that order.
- Adding to a definite integral answer, which is wrong because definite integrals evaluate to a number, while is only used for indefinite integrals.
Practice Questions
- 1 Let . Find and then find .
- 2 Evaluate using an antiderivative.
- 3 A function is positive and increasing on . Explain what the Fundamental Theorem says about the slope of .