FTC & Net Change Lab

Investigate the Fundamental Theorem of Calculus in three modes. Build the accumulation function and verify that its derivative equals the integrand, evaluate definite integrals using antiderivatives, and compare displacement with total distance for particle motion problems.

Guided Experiment: Building F(x) from f(x)

If F(x) is defined as the integral of f(t) from a to x, what is the relationship between F'(x) and f(x)? How does the accumulated area change as x increases?

Write your hypothesis in the Lab Report panel, then click Next.

f(t) = t

-1-0.500.511.522.533.54-2-1012345a = 0x = 3

Controls

Accumulation Function

Definition
Current Value
FTC Part 1 Verification
F'(x) = f(x) confirmed
Known Antiderivative
F(x) = x²/2

Data Table

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#TabFunctionBoundsResultNote
0 / 500
0 / 500
0 / 500

Reference Guide

FTC Part 1

If f is continuous on [a, b] and F is defined by the accumulation integral, then F is differentiable and its derivative is the original function.

F(x)=∫axf(t) dt  ⟹  F′(x)=f(x)F(x) = \int_a^x f(t)\,dt \implies F'(x) = f(x)

This says that integration and differentiation are inverse processes. The rate of change of accumulated area equals the height of the function.

FTC Part 2

If F is any antiderivative of f on [a, b], then the definite integral equals the net change in F.

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)

This provides an efficient way to compute definite integrals. Instead of taking limits of Riemann sums, find an antiderivative and evaluate at the endpoints.

Net Change Theorem

The integral of a rate of change over an interval gives the total (net) change in the quantity.

∫abF′(x) dx=F(b)−F(a)\int_a^b F'(x)\,dx = F(b) - F(a)

If F'(x) represents velocity, then the integral gives the net displacement. If it represents a population growth rate, the integral gives the total population change.

Displacement vs Distance

Displacement accounts for direction while total distance does not.

Displacement=∫abv(t) dt\text{Displacement} = \int_a^b v(t)\,dt
Total Distance=∫ab∣v(t)∣ dt\text{Total Distance} = \int_a^b |v(t)|\,dt

When velocity changes sign, positive and negative regions partially cancel in the displacement integral but add in the distance integral.