AP Calc Fundamental Theorem of Calculus, Step by Step

The Fundamental Theorem of Calculus (FTC) connects derivatives and integrals. Part 1 says that if you build a function by accumulating area under f from a fixed point to x, its derivative is f(x). Part 2 says you can evaluate a definite integral by finding an antiderivative F and computing F(b) − F(a).

Both parts are central to AP Calculus AB and BC. They appear in the integration unit, and they come back throughout the free-response section in accumulation and net change problems.

What does the Fundamental Theorem of Calculus say?

Both parts require f to be continuous on the interval. On AP questions this is almost always given, but it is worth remembering why the condition is there.

FTC Part 1: derivatives of accumulation functions

An accumulation function adds up signed area under f from a fixed starting point to a variable endpoint x. Part 1 says the rate at which that area grows is the height of f at x.

Example 1: the basic case

Find d/dx of ∫ from 2 to x of (t² + 1) dt.

By FTC Part 1, replace t with x: the answer is x² + 1. No integration needed.

Example 2: a function as the upper limit

Find d/dx of ∫ from 0 to x³ of cos(t) dt.

When the upper limit is a function g(x), use the chain rule: plug g(x) into f, then multiply by g′(x).

• f(g(x)) = cos(x³)
• g′(x) = 3x²
• Answer: 3x² cos(x³)

Example 3: x in the lower limit

Find d/dx of ∫ from x to 5 of √t dt.

Flip the limits, which changes the sign: this equals −∫ from 5 to x of √t dt. The derivative is −√x.

Example 4: x in both limits

Find d/dx of ∫ from x to x² of e^t dt.

Split at a constant, then apply Part 1 to each piece. The general rule is (upper function plugged in) times (its derivative) minus (lower function plugged in) times (its derivative):

2x · e^(x²) − e^x

FTC Part 1 rules at a glance

FTC Part 2: evaluating definite integrals

Part 2 is the version you use most often. Find any antiderivative, evaluate it at the upper limit, and subtract its value at the lower limit.

Example 5: a polynomial

Evaluate ∫ from 1 to 3 of (2x + 1) dx.

1. An antiderivative is F(x) = x² + x.
2. F(3) = 9 + 3 = 12.
3. F(1) = 1 + 1 = 2.
4. The integral is 12 − 2 = 10.

You do not need a + C here, because it cancels in F(b) − F(a).

The net change theorem: FTC in context

On the AP exam, Part 2 is most often tested as the net change theorem: the integral of a rate of change gives the total change.

f(b) = f(a) + ∫ from a to b of f′(x) dx

In words: ending amount = starting amount + accumulated change.

Example 6: using a known value

Suppose f(2) = 5 and ∫ from 2 to 6 of f′(x) dx = 7. Find f(6).

f(6) = f(2) + 7 = 12.

Example 7: a tank problem

A tank holds 50 gallons at time t = 0. Water flows in at r(t) = 3t gallons per minute. How much water is in the tank at t = 10 minutes?

1. Accumulated change: ∫ from 0 to 10 of 3t dt = (3/2)(10²) − 0 = 150 gallons.
2. Amount at t = 10: 50 + 150 = 200 gallons.

This structure, starting amount plus an integral of a rate, is one of the most common setups on AP Calculus free-response questions. Many of these questions are on the calculator-active part of the exam, where you set up the integral and let the calculator evaluate it.

Accumulation functions from a graph

A classic AP free-response format gives you the graph of f and defines g(x) = ∫ from a to x of f(t) dt. Then it asks about g.

The FTC turns these into graph-reading questions:

• g(x) is the signed area under f from a to x. Find it with geometry: rectangles, triangles, and semicircles.
• g′(x) = f(x). Read it straight off the graph.
• g″(x) = f′(x). It is the slope of the graph of f.
• g is increasing where f is positive, and decreasing where f is negative.
• g has a relative max where f changes from positive to negative.
• g is concave up where f is increasing.

The trap is mixing up which function you are describing. Write "g′(x) = f(x)" at the top of your work so you keep the roles straight.

Common FTC mistakes on the AP exam

• Forgetting the chain rule when the upper limit is a function of x.
• Forgetting the negative sign when x is in the lower limit.
• Ignoring signed area. Area below the x-axis counts as negative in an accumulation function.
• Leaving out the starting value in net change problems. The integral gives the change, not the final amount.
• Weak justifications. When explaining where g has a maximum, refer to f: "g has a relative maximum at x = 3 because g′ = f changes from positive to negative there."

Our AP Calculus FRQ tips explain exactly how readers score justifications like these.

Where the FTC fits in the AP Calc course

The FTC sits in the integration and accumulation unit, which carries significant exam weight in both AB and BC. To see how it connects to the rest of the course, read AP Calculus AB units explained. For a list of the integral rules you will use alongside the FTC, see our AP Calculus formula sheet.

When you start taking full practice exams, the AP Calculus AB score calculator will give you an estimate of where you stand. Score cutoffs vary by year, so treat projections as a guide.

How Ivystar can help

The FTC is where many students realize calculus is a connected story rather than a list of rules, and a good tutor makes that connection click. Ivystar's AP Exam Prep program offers 1-on-1 or small-group sessions, and our students average a one-letter-grade improvement in the class. Book a free strategy call to talk through your goals.