What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is the statement that differentiation and integration are inverse operations. It has two parts: the first says that the derivative of an accumulator function recovers the integrand, and the second says that a definite integral can be evaluated by finding any antiderivative and subtracting its values at the two endpoints.

What does F prime of x equals f of x mean geometrically?

It means the slope of the accumulator curve F at any point x is exactly the height of the integrand curve f at that same x. Where f is positive, F climbs. Where f is zero, F is flat. Where f is large, F climbs steeply. The two curves are linked: one is the running area, the other is the rate at which that area is being added.

How do you compute a definite integral using the FTC?

Find any antiderivative G of the integrand f, then evaluate G at the upper bound and subtract its value at the lower bound. The result is the definite integral from the lower bound to the upper bound. For example, to integrate t squared from zero to two, take the antiderivative t cubed over three and compute eight over three minus zero, which gives eight over three.

What is the difference between Part 1 and Part 2 of the FTC?

Part 1 builds a new function as the accumulated integral and proves that its derivative is the original integrand. Part 2 goes the other direction: it uses Part 1 to compute the value of a specific definite integral by evaluating any antiderivative at the two endpoints. Part 1 is the theoretical link; Part 2 is the practical evaluation tool.

Why is the FTC important?

Without the FTC, computing a definite integral would require summing infinitely many thin rectangles, which is impractical. The FTC reduces this to finding a single antiderivative and evaluating at two points. It also explains why integration and differentiation are studied together: they are not separate topics but two sides of one operation.

Fundamental Theorem of Calculus

Quadraticf(t) = t²F(x) = x³/3 − a³/3

880 × 460

f(t) = t² (integrand)F(x) = ∫ₐˣ f(t) dt (accumulator)shaded area

At the moving point xx = 2 (a = 0)

Area ∫ₐᵝ f(t) dt

2.667

shaded region under f

F(x)

2.667

height on F

f(x) = F'(x)

4

slope of F at x

The shaded area from a to x equals the height of F at x; and the slope of F at x equals f(x).FTC

Applieda=0x=2F'(x) = f(x)

Quadratic: accumulator and integrand

The shaded region is the area under f from a to x. Sliding x grows or shrinks that area, and the height of F tracks it exactly.

Integrand ·

f(t) = t²

Accumulator ·

F(x) = x³/3 − a³/3

Right now

a =

0

, x =

2

. Shaded area =

2.667

, F(x) =

2.667

(they match by construction). f(x) =

4

, which is also

F'(x)

— the slope of the F curve at x.

The two halves of the theorem

Part 1 (the link). Define

F(x) = \int_a^x f(t)\,dt

— the accumulated area. Then F is differentiable and

F'(x) = f(x)

. Differentiation undoes integration.

Part 2 (evaluation). If G is any antiderivative of f, then

\int_a^b f(t)\,dt = G(b) - G(a)

. That's how you compute definite integrals without summing rectangles forever.

Key Terms

Getting Started

The Function Families

The a and x Sliders

The Shaded Area

Reading the At-The-Point Card

Display Toggles

What Is the FTC

Part 1 - The Accumulator

Part 2 - Evaluation

Related Concepts

Key Terms

Fundamental Theorem of Calculus (FTC) — the statement that differentiation and integration are inverse operations. Has two parts: Part 1 says the derivative of an accumulator function equals the integrand; Part 2 gives a way to compute definite integrals using antiderivatives.

Integrand — the function

f(t)

being integrated. The curve whose area you are accumulating.

Accumulator function — the function

F(x) = \int_a^x f(t)\, dt

. It returns the signed area under

f

from a fixed lower bound

a

to the moving upper bound

x

.

Antiderivative — any function

G

with

G'(x) = f(x)

. Antiderivatives differ by a constant; the accumulator is one specific antiderivative pinned by

F(a) = 0

.

Definite integral — the signed area

\int_a^b f(t)\, dt

. Computed via Part 2 as

G(b) - G(a)

for any antiderivative

G

.

Signed area — area counted positively where

f > 0

and negatively where

f < 0

Getting Started

The page opens with the Quadratic family

f(t) = t^2

loaded, the lower bound

a

parked at

0

, and the upper bound

x

2

. You see three things together on the graph:

• The solid blue curve of

f

(the integrand).

• The dashed deep-blue curve of

F

(the accumulator).

• The shaded region between

f

and the t-axis from

a

x

.

Two thin vertical lines mark

a

and

x

on the t-axis. Marker dots sit on

f

and

F

at the upper bound

x

.

The boxed card below the graph displays three numbers: the shaded area, the height

F(x)

, and the value

f(x) = F'(x)

. The first two always match. The third is the slope of

F

x

.

To explore quickly, drag the upper bound x slider and watch every part of the picture update at once.

The Function Families

Six families are organized into two groups in the left panel, each shipped with a closed-form antiderivative.

Polynomial:

• Identity —

f(t) = t

F(x) = x^2/2 - a^2/2

. The simplest case where signed area depends on whether

x > a

x < a

.

• Quadratic —

f(t) = t^2

F(x) = x^3/3 - a^3/3

. The power rule in its cleanest form.

• Cubic —

f(t) = t^3

F(x) = x^4/4 - a^4/4

. Odd integrand; useful for seeing how negative regions cancel positive ones.

Transcendental:

• Sine —

f(t) = \sin t

F(x) = -\cos x + \cos a

. Periodic accumulator with bounded swings.

• Cosine —

f(t) = \cos t

F(x) = \sin x - \sin a

. Same period; accumulator is a sine.

• Exponential —

f(t) = e^t

F(x) = e^x - e^a

. The accumulator grows the same way the integrand does.

The a and x Sliders

The lower bound a slider sets the fixed starting point of integration. By construction

F(a) = 0

— the accumulator passes through zero exactly at

x = a

. Move

a

and the whole

F

curve shifts vertically; the integrand

f

is unaffected.

The upper bound x slider sweeps the moving endpoint:

• The right edge of the shaded region tracks

x

in real time.

• The marker dots on both curves slide to the new

x

.

• The numeric card updates with the new area, the new

F(x)

, and the new

f(x)

.

Try setting

a = 0

and dragging

x

from

0

rightward. The shaded area grows as

x

grows, and the height of the

F

curve climbs by exactly the same amount.

The Reset button next to the Parameters label restores the family's default

a

and

x

The Shaded Area

The shaded region between the integrand

f

and the t-axis from

a

x

is the visual heart of the tool. Its signed area is what the definite integral

\int_a^x f(t)\, dt

computes.

A few things to notice as you slide

x

:

• When

f > 0

on the interval, the shaded area is positive and grows.

• When

f < 0

, the shaded area is negative — the integrand sits below the t-axis, and the integral subtracts that contribution.

• When the integrand crosses zero, the accumulator

F

momentarily stops growing. A flat spot of

F

— and a local maximum or minimum — sits exactly there.

• When

a = x

, the shaded region collapses to a single point. Area is zero;

F(a) = 0

by definition.

• Sliding

x

to the left of

a

flips the orientation, and the area enters with the opposite sign.

Reading the At-The-Point Card

The boxed card below the graph displays the three quantities at the moving point

x

:

• Area — the signed area of the shaded region from

a

x

. Computed directly as

\int_a^x f(t)\, dt

.

• F(x) — the height of the accumulator curve at

x

. Computed from the closed-form antiderivative.

• f(x) = F'(x) — the value of the integrand at

x

, equal to the slope of

F

x

.

The first two numbers are always equal by construction — that's how

F

is defined.

The third number is the punchline of Part 1 of the FTC: the slope of the accumulator at

x

is exactly the integrand evaluated at

x

. Differentiation undoes integration.

Read the row as you slide: left tells you the area piling up, middle tells you the same area as a height, right tells you how fast that pile is growing.

Display Toggles

The Display section in the left panel hides individual layers when one is in the way:

• f(t) — toggles the integrand curve. Off, you see just the accumulator and the shaded area outline.

• F(x) — toggles the dashed accumulator curve. Off, the picture reduces to a classical *area under f* view without the FTC overlay.

• area — toggles the shaded fill. Off, the two function curves remain but the area visualization is gone.

The legend below the graph updates to show only the visible layers.

The Accent color picker at the bottom recolors the highlight throughout — slider track, verdict border, key labels — useful for screenshots or personal preference.

What Is the FTC

The Fundamental Theorem of Calculus is the bridge between the two central operations of calculus: differentiation (slope of a curve) and integration (area under a curve). It says the two operations are *inverses* of each other.

The theorem has two parts, both essential.

Part 1 establishes that if you define a function by integrating

f

from a fixed lower bound up to a moving upper bound, the derivative of that function recovers

f

. Symbolically, if

F(x) = \int_a^x f(t)\, dt

, then

F'(x) = f(x)

.

Part 2 uses Part 1 to give a practical way to compute definite integrals: pick *any* antiderivative

G

f

, and the integral

\int_a^b f(t)\, dt

equals

G(b) - G(a)

.

For full coverage of proofs and applications, see the fundamental theorem of calculus page.

Part 1 - The Accumulator

FTC Part 1. If

f

is continuous on

[a, b]

and

F(x) = \int_a^x f(t)\, dt

for

x

[a, b]

, then

F

is differentiable on

(a, b)

and

F'(x) = f(x).

The intuition: push

x

a little further right by

\Delta x

. The shaded area grows by a thin strip of width

\Delta x

and height roughly

f(x)

. So

\Delta F \approx f(x) \cdot \Delta x

, which means

F'(x) = f(x)

in the limit.

This is the reason the accumulator is sometimes called the *integral function* — it shows that integration produces a function whose derivative is the original integrand. Differentiation undoes integration, point by point.

In the tool, you can verify this visually. Pick any

x

, note the slope of the

F

curve there (use the rate at which the dot is climbing), and compare it to the height of

f

at the same

x

. They match.

Part 2 - Evaluation

FTC Part 2. If

f

is continuous on

[a, b]

and

G

is *any* antiderivative of

f

(so

G'(x) = f(x)

everywhere on the interval), then

\int_a^b f(t)\, dt = G(b) - G(a).

This is the practical workhorse of integration. Without it, computing a definite integral would mean summing the areas of infinitely many thin rectangles. With it, you find an antiderivative — usually by reversing a differentiation rule — and evaluate at the endpoints.

For example, to compute

\int_0^2 t^2\, dt

:

• Find an antiderivative:

G(t) = t^3/3

works.

• Evaluate at the endpoints:

G(2) - G(0) = 8/3 - 0 = 8/3

.

That's exactly what the tool reports when you load the Quadratic family with

a = 0

and

x = 2

.

For deeper coverage of integration techniques and antiderivative tables, see the integration techniques page.

Related Concepts

Definite integrals — the area under a curve between two specific bounds. The output of the FTC's Part 2 evaluation.

Indefinite integrals — the family of antiderivatives, written

\int f(x)\, dx = G(x) + C

. The constant

C

disappears in any definite integral.

Riemann sums — the construction that defines the integral as a limit of rectangle-area sums. The FTC turns this limit into a finite computation. See the Riemann sum visualizer.

Antiderivative rules — power rule, exponential rule, trig integrals. The reverse direction of the differentiation rules.

Substitution and integration by parts — techniques for finding antiderivatives of more complicated integrands.

Improper integrals — integrals where the bounds are infinite or the integrand is unbounded. The FTC still applies after a limiting process.

Derivatives — the inverse operation. The FTC is the explicit statement of that inverse relationship.

Visual tools for calculus — other interactive visualizers covering limits, continuity, derivatives, and Riemann sums.