The Fundamental Theorem of Calculus, Part 1

What the Fundamental Theorem of Calculus, Part 1 Is About

The Fundamental Theorem of Calculus is one of the most important ideas in all of calculus because it connects the two major operations you've been studying: differentiation and integration.

Before this theorem, you can think of an integral such as as describing accumulated area or accumulated change. The Fundamental Theorem of Calculus, Part 1 tells you something remarkable: if you build a function by accumulating the values of another function, then differentiating that accumulation function gives you the original function back.

Suppose we define an accumulation function by

Then, under the appropriate continuity conditions, the derivative of is simply the function inside the integral:

This is the central idea of FTC Part 1. The integral accumulates change, while the derivative tells you the instantaneous rate at which that accumulation is changing.

The Fundamental Theorem of Calculus, Part 1

If is continuous on an interval containing , and

then

In words:

The lower limit of integration is fixed, while the upper limit is the variable . That changing upper limit is what makes an accumulation function.

Why This Makes Sense

Imagine that represents the rate at which water enters a tank. Define as the total amount of water that has entered the tank between time and time .

If you increase by a very small amount, the additional amount of water is approximately the current rate multiplied by the small change in time:

Therefore, the instantaneous rate of change of the accumulated amount is exactly the current value of the rate function:

That is the intuition behind FTC Part 1: the derivative of accumulated change is the original rate of change.

The Basic Pattern to Recognize

Whenever you see a function of the form

you should immediately think:

The integral disappears when you differentiate because FTC Part 1 says that differentiation reverses this accumulation process.

Worked Examples

FTC Part 1 vs. FTC Part 2

The two parts of the Fundamental Theorem of Calculus are related, but they answer different questions.

Part 1 is primarily about differentiating an accumulation function. If the variable is inside a limit of integration, FTC Part 1 is probably the theorem you need.

Part 2 is about evaluating a definite integral using an antiderivative. It gives the familiar rule

A useful AP Calculus distinction is:

• FTC Part 1: "I have an integral-defined function. What is its derivative?"
• FTC Part 2: "I have a definite integral. How can I calculate its exact value?"

Common Mistakes

Practice Problems