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Exploring
Accumulations of Change

Understand how definite integrals capture the total accumulation of a quantity, connecting rates of change to net change through area under a curve.

AP Calculus AB

What Is Accumulation?

In calculus, accumulation is the process of adding up infinitely many small changes to find a total change. If you know a rate of change, such as velocity, you can accumulate it over time to find the total displacement.

Rate
How fast something changes: velocity, flow rate, growth rate
Accumulation
The total change: distance traveled, volume filled, population gained

Follow the salt through a working mixing rig

A stirred transparent tank has one upper inlet and one lower outlet with a valve. The exact diagram links its 100 L salt balance to a computed approach to equilibrium.For net flow R(t) = 4t − t² gal/min and 10 gal initially, V(t) = 10 + 2t² − t³/3. Volume peaks at t = 4 minutes, when the net rate changes from positive to negative.

Signed net flow and inventory

For net flow R(t) = 4t − t² gal/min and 10 gal initially, V(t) = 10 + 2t² − t³/3. Volume peaks at t = 4 minutes, when the net rate changes from positive to negative.For net flow R(t) = 4t − t² gal/min and 10 gal initially, V(t) = 10 + 2t² − t³/3. Volume peaks at t = 4 minutes, when the net rate changes from positive to negative.

Visualizing Area Under a Curve

The accumulation of \( f(x) \) from \( a \) to \( b \) equals the signed area between the graph and the \( x \)-axis. Area above the axis is positive; area below is negative.

\(x\)
\(y\)
\(a\)
\(b\)
 Area = Accumulation
\(f(x)\)

From Riemann Sums to Integrals

A Riemann sum approximates accumulation by dividing the interval into \( n \) sub-intervals and summing rectangular areas. As \( n \to \infty \), the sum becomes the definite integral.

\(\Delta x\)
\( \displaystyle\sum_{i=1}^n f(x_i) \Delta x \xrightarrow{n\to\infty} \int_a^b f(x) \, dx \)

The Definite Integral

\( a \) and \( b \)
Lower and upper limits of integration: the interval boundaries
\( f(x) \)
The integrand: the function being accumulated
\( dx \)
Infinitesimal width: indicates the variable of integration

The Fundamental Theorem of Calculus

The FTC connects differentiation and integration, providing a powerful tool to evaluate definite integrals without computing limits of Riemann sums.

Accumulation Functions

An accumulation function tracks how much total change has occurred from a fixed starting point \( a \) up to a variable endpoint \( x \):

\(t\)
\(y\)
\(a\)
\(x\)
\(F(x)\)
\(f(t)\)

Definite vs Indefinite Integrals

Definite Integral Indefinite Integral
Notation \( \int_a^b f(x)\,dx \) \( \int f(x)\,dx \)
Result A number: net accumulation A function: family of antiderivatives
Bounds Has limits \( a \) and \( b \) No limits of integration
Constant No \( +C \) Includes \( +C \)
Interpretation Signed area under curve General antiderivative

Understanding Signed Area

When \( f(x) < 0 \) on part of the interval, the integral counts that region as negative. The definite integral gives net accumulation, not total area.

\(x\)
 + Area − Area
\(a\)
\(b\)

How to Evaluate a Definite Integral

1
Find the antiderivative. Determine \( F(x) \) such that \( F'(x) = f(x) \).
2
Apply the bounds. Compute \( F(b) - F(a) \) using the Fundamental Theorem (FTC Part 2).
3
Simplify. The result is a single numerical value representing the net accumulation.
4
Interpret. Relate the value back to the context: distance, volume, total change, etc.

Example 1: Basic Polynomial

Evaluate \( \displaystyle\int_1^3 (2x + 1)\,dx \).

1
Antiderivative: \( F(x) = x^2 + x \)
2
Apply bounds: \( F(3) - F(1) = (3^2 + 3) - (1^2 + 1) = 12 - 2 \)
3
Result: \( \displaystyle\int_1^3 (2x + 1)\,dx = 10 \)
\(x\)
\(y\)
 1 3 = 10

Example 2: Trigonometric Integral

Evaluate \( \displaystyle\int_0^{\pi} \sin(x)\,dx \).

1
Antiderivative: \( F(x) = -\cos(x) \)
2
Apply bounds: \( F(\pi) - F(0) = -\cos(\pi) - (-\cos(0)) = -(-1) - (-1) = 1 + 1 \)
3
Result: \( \displaystyle\int_0^{\pi} \sin(x)\,dx = 2 \)
\(0\)
\(\pi\)
\(\sin(x)\)
 = 2

Example 3: Velocity and Displacement

A particle moves with velocity \( v(t) = t^2 - 4t + 3 \) m/s. Find the net displacement from \( t = 0 \) to \( t = 4 \).

1
Set up: \( \displaystyle\int_0^4 (t^2 - 4t + 3)\,dt \)
2
Antiderivative: \( F(t) = \frac{t^3}{3} - 2t^2 + 3t \)
3
Evaluate: \( F(4) - F(0) = \left(\frac{64}{3} - 32 + 12\right) - 0 = \frac{64}{3} - 20 = \frac{4}{3} \)

Properties of Definite Integrals

Linearity
\( \int_a^b [cf(x) + g(x)]\,dx = c\int_a^b f(x)\,dx + \int_a^b g(x)\,dx \)
Additivity
\( \int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx \)
Reversal
\( \int_b^a f(x)\,dx = -\int_a^b f(x)\,dx \)

Common Mistakes to Avoid

✕ Forgetting \( +C \) Context
Adding \( +C \) to a definite integral. The constant cancels: \( (F(b) + C) - (F(a) + C) = F(b) - F(a) \).
✕ Ignoring Sign
Treating all area as positive. The definite integral gives net (signed) accumulation, not total area.
✕ Swapping Bounds
Computing \( F(a) - F(b) \) instead of \( F(b) - F(a) \). The order matters and flipping it negates the result.

Question 1 of 5

Evaluate \( \displaystyle\int_0^2 3x^2\,dx \).

Question 2 of 5

If \( \displaystyle\int_1^5 f(x)\,dx = 7 \) and \( \displaystyle\int_5^8 f(x)\,dx = -3 \), what is \( \displaystyle\int_1^8 f(x)\,dx \)?

Question 3 of 5

What does \( \displaystyle\int_a^b v(t)\,dt \) represent when \( v(t) \) is the velocity of a particle?

Question 4 of 5

If \( F(x) = \displaystyle\int_2^x (3t^2 + 1)\,dt \), what is \( F'(x) \)?

Question 5 of 5

What is \( \displaystyle\int_3^3 (x^4 + 7x - 2)\,dx \)?