The First Fundamental Theorem of Calculus connects two big ideas: accumulation and rate of change. It says that when a function F(x) is built by adding up signed area under another function f(t), the derivative of F gives back the original function f. This matters because it explains why derivatives and integrals are inverse processes in a precise way.
It also gives a powerful method for understanding changing quantities from their accumulated totals.
Understanding Calculus: The Fundamental Theorem Part 1
A useful way to understand the theorem is to focus on a very small change in the endpoint. Suppose the endpoint moves from x to x plus a tiny amount h. The accumulated quantity changes only by the thin strip between those two positions.
If the graph is smooth near x, that strip has height close to the function value at x and width h. Its contribution is therefore close to the function value times h. Dividing the change in accumulation by h gives a value close to the function value.
As h becomes extremely small, the approximation becomes exact. This is why continuity matters. A jump or sharp break can make the local strip harder to describe, even though an integral may still exist.
Signed area is really a bookkeeping system for net change. Parts of a graph above the horizontal axis add to the running total. Parts below it remove from that total.
This means an accumulated function can return to zero after having changed a great deal. For example, water flowing into a tank for one hour and out for the next hour can leave the tank at its original level. The net change is zero, but water still moved in both directions.
Students often confuse net area with total physical area. Total area treats every region as positive. Net area keeps track of direction, which is usually the more useful idea in science.
The theorem becomes especially important when the endpoint is controlled by another changing quantity. Imagine that a sensor records a rate as time passes, but the time used in the integral is itself determined by the position of a moving object. The rate of the accumulated result depends on two things.
It depends on the rate shown by the graph at the current endpoint. It depends on how fast the endpoint is moving. If the endpoint moves twice as fast, it passes through twice as much of the accumulated region each second.
This is the chain rule in a physical setting. It appears in problems about filling containers, electric charge, distance traveled, population change, and heat transfer.
When working with these problems, first identify what is being accumulated and its units. If a graph gives velocity in meters per second, its accumulation has units of meters. If a graph gives electric current in amperes, its accumulation has units of charge.
Units provide a quick error check. Next, notice whether the lower endpoint is fixed or changing. A fixed starting point contributes no new change.
A changing starting point does contribute, with an opposite effect because it removes part of the interval. Finally, keep the integration variable separate from the input variable.
The letter inside the integral is only a placeholder that runs across the interval. Changing that letter does not change the result, but confusing it with the moving endpoint can lead to incorrect derivatives.
Key Facts
- If F(x) = ∫_a^x f(t) dt, then F'(x) = f(x).
- The variable t is a dummy variable of integration, while x controls the moving upper limit.
- F(x) measures signed area from t = a to t = x under the curve y = f(t).
- If f(x) > 0, then F is increasing at x because F'(x) > 0.
- If f(x) < 0, then F is decreasing at x because F'(x) < 0.
- For G(x) = ∫_a^{g(x)} f(t) dt, the chain rule gives G'(x) = f(g(x))g'(x).
Vocabulary
- Accumulation function
- A function defined by an integral with a variable limit that measures the net amount accumulated from a starting point.
- Signed area
- Area counted as positive when the graph is above the axis and negative when the graph is below the axis.
- Dummy variable
- A placeholder variable inside an integral that can be renamed without changing the value of the integral.
- Upper limit of integration
- The endpoint of an integral that can move and determine how much area is included.
- First Fundamental Theorem of Calculus
- The theorem stating that the derivative of an accumulation function F(x) = ∫_a^x f(t) dt is the integrand evaluated at x.
Common Mistakes to Avoid
- Forgetting that ∫_a^x f(t) dt is a function of x. The upper limit moves, so the accumulated area changes as x changes.
- Treating t and x as the same variable. The variable t is only used inside the integral, while x sets the moving endpoint.
- Writing F'(x) = ∫_a^x f'(t) dt. The theorem says differentiating the accumulation function gives f(x), not the integral of the derivative.
- Ignoring negative area below the axis. Area below the t-axis decreases the accumulation function because the integral uses signed area.
Practice Questions
- 1 Let F(x) = ∫_2^x (3t^2 - 4t) dt. Find F'(x) and then evaluate F'(3).
- 2 Let G(x) = ∫_1^{x^2} sqrt(1 + t^3) dt. Use the First Fundamental Theorem and the chain rule to find G'(x).
- 3 Suppose F(x) = ∫_0^x f(t) dt and the graph of f is below the t-axis on the interval 1 < x < 4. Explain what happens to F on that interval and why.