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Accumulation Functions
and Area

Interpret how accumulation functions built from definite integrals encode the signed area under a curve, and connect their behavior to the original function.

AP Calculus BC

What Is an Accumulation Function?

An accumulation function is defined as a definite integral with a variable upper limit:

As \( x \) changes, \( F(x) \) accumulates the signed area between the graph of \( f(t) \) and the \( t \)-axis from \( t = a \) to \( t = x \). When \( f(t) > 0 \), area is added; when \( f(t) < 0 \), area is subtracted.

Visualizing Signed Area

t f(t) a c x + Area - Area
\(F(x)\) = net area

Why Accumulation Matters

Physics
Velocity integrated over time gives displacement: the net change in position.
Economics
A marginal cost function integrated gives the total variable cost from producing units.
Biology
A rate of growth integrated gives total population change over an interval.

In each case, the accumulation function translates a rate into a total quantity: exactly what \( F(x)=\int_a^x f(t)\,dt\) does.

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.Suppose a tank receives and releases 4 gal/min for 5 minutes. Its 200 gal inventory stays constant while 20 gal enters and 20 gal leaves: net change is zero and total throughput is 40 gal.

Net inventory versus throughput

Suppose a tank receives and releases 4 gal/min for 5 minutes. Its 200 gal inventory stays constant while 20 gal enters and 20 gal leaves: net change is zero and total throughput is 40 gal.Suppose a tank receives and releases 4 gal/min for 5 minutes. Its 200 gal inventory stays constant while 20 gal enters and 20 gal leaves: net change is zero and total throughput is 40 gal.

The Fundamental Theorem (Part 1)

This is the bridge: the derivative of the accumulation function equals the original integrand evaluated at the upper limit. The rate at which area accumulates at \( x \) is exactly \( f(x) \).

FTC with the Chain Rule

When the upper limit is a function \( g(x) \) instead of just \( x \), we apply the chain rule:

1
Evaluate the integrand at the upper limit: plug \( g(x) \) into \( f \).
2
Multiply by the derivative of the upper limit: \( g'(x) \).
3
If the lower limit is also a function \( h(x) \), subtract \( f\!\bigl(h(x)\bigr)\cdot h'(x) \).

Increasing vs. Concave Up

F is increasing F is concave up
Condition \( F'(x) = f(x) > 0 \) \( F''(x) = f'(x) > 0 \)
Read from graph of \( f \) \( f \) is above the \( t \)-axis \( f \) is increasing
Meaning for area Net area is growing Area is growing at a faster rate

How to Analyze \( F(x) \)

1
Identify \( f \). The integrand is \( F'(x) \). Every question about \( F \) starts from the graph or formula of \( f \).
2
Sign of \( f \). Where \( f > 0 \), \( F \) increases. Where \( f < 0 \), \( F \) decreases.
3
Zeros of \( f \). Where \( f \) crosses the axis, \( F \) has a critical point. Check sign change for max/min.
4
Slope of \( f \). Where \( f \) is increasing, \( F \) is concave up. Where \( f \) is decreasing, \( F \) is concave down.
5
Compute values. Use geometry or the integral to find \( F(x) \) at specific points (areas of triangles, rectangles, semicircles).

Example 1: Reading \( f \) to Sketch \( F \)

Given the graph of \( f(t) \) below, with \(\displaystyle F(x) = \int_0^x f(t)\,dt\), describe the behavior of \( F \).

t f(t) 1 2 3 4 5
\(f > 0\): \(F\) increases
\(f < 0\): \(F\) decreases
\(F\) has max

Example 1: Computing \( F \) Values

Using the same piecewise-linear \( f(t) \) from the previous slide, compute \( F(3) \) and \( F(5) \).

1
\( F(3) = \int_0^3 f(t)\,dt \). The region from \( t=0 \) to \( t=3 \) consists of a rectangle on \([0,2]\) and a triangle on \([2,3]\).
2
\( F(3) = (2)(2) + \frac{1}{2}(1)(2) = 4 + 1 = \) 5.
3
\( F(5) = F(3) + \int_3^5 f(t)\,dt \). The region from \(t=3\) to \(t=5\) is a triangle below the axis with area \( \frac{1}{2}(2)(2) = 2 \). Since it is below the axis, its signed contribution is -2.
4
\( F(5) = 5 + (-2) = \) 3.

Example 2: Concavity of \( F \)

t f(t) 1 2 3 4
\(f\) increasing
\(F\) concave up
\(f\) has max
\(F\) inflection pt
\(f\) decreasing
\(F\) concave down

Example 3: Chain Rule in Action

Find \(\displaystyle \frac{d}{dx}\int_1^{x^2} \sqrt{t^3+1}\,dt\).

1
Identify the pieces. Here \( f(t) = \sqrt{t^3+1} \) and the upper limit is \( g(x) = x^2 \).
2
Apply FTC + Chain Rule: evaluate \( f \) at \( g(x) \): \( \sqrt{(x^2)^3 + 1} = \sqrt{x^6+1} \).
3
Multiply by \( g'(x) = 2x \).

Example 4: Full Analysis from a Graph

The graph of \( f \) consists of line segments. Let \(\displaystyle F(x) = \int_0^x f(t)\,dt\). Find where \( F \) has a relative minimum.

t f(t) 1 2 3 4 5 1 -1
\(F\) has rel. min
\(f\) changes: - to +

Common Mistakes to Avoid

✕ Confusing \( f \) and \( F \)
A zero of \( f \) is a critical point of \( F \), not a zero of \( F \). Don't confuse where the integrand is zero with where the accumulation is zero.
✕ Forgetting the Chain Rule
When the upper limit is \( g(x) \), you must multiply by \( g'(x) \). Omitting this factor is one of the most common FTC errors.
✕ Ignoring Signed Area
Area below the axis contributes negatively. Don't add all areas as positive: \( F \) tracks net (signed) area.

Summary: Reading \( f \) for \( F \)

If \( f(x) \) is... Then \( F(x) \) is... Because...
Positive Increasing \( F' = f > 0 \)
Negative Decreasing \( F' = f < 0 \)
Zero (sign change + to -) Local max \( F' \) changes + to -
Zero (sign change - to +) Local min \( F' \) changes - to +
Increasing Concave up \( F'' = f' > 0 \)
Decreasing Concave down \( F'' = f' < 0 \)
Local max or min Inflection point \( F'' = f' \) changes sign

Question 1 of 5

Let \(\displaystyle F(x) = \int_0^x f(t)\,dt\). If \( f(x) > 0 \) and \( f \) is decreasing on \((2, 5)\), which statement about \( F \) is true on \((2, 5)\)?

Question 2 of 5

What is \(\displaystyle \frac{d}{dx}\int_2^{x^3} \cos(t)\,dt\)?

Question 3 of 5

Let \(\displaystyle G(x)=\int_0^x f(t)\,dt\). If \( f \) crosses the \( t \)-axis from positive to negative at \( t = 4 \), what happens to \( G \) at \( x = 4 \)?

Question 4 of 5

The graph of \( f \) on \([0, 6]\) consists of a semicircle of radius 2 above the axis from \( t = 0 \) to \( t = 4 \) and a triangle with base 2 and height 3 below the axis from \( t = 4 \) to \( t = 6 \). What is \(\displaystyle \int_0^6 f(t)\,dt\)?

Question 5 of 5

Let \(\displaystyle H(x) = \int_1^x (3t^2 - 6t)\,dt\). At what value of \( x \) does \( H \) have a relative minimum for \( x > 0 \)?