6.1 Integration and Accumulation of Change

Accumulation of change is the total amount something changes over an interval, and you find it by computing the area between a rate of change graph and the x-axis. When the shape under the curve is simple, you can use geometry; when it is not, you can approximate with rectangles (Riemann sums) before learning exact integration later in the unit. For AP Calculus, include units because accumulated change combines rate units with the input units.

Why This Matters for the AP Calculus Exam

This is the foundation for all of Unit 6, which carries a large share of the exam. Once you see that area under a rate graph equals accumulated change, the rest of integration starts to make sense. On both multiple-choice and free-response questions, you will read graphs of rates (like velocity, flow rate, or population growth rate), find accumulated change using geometry or approximations, and explain what your answer means in context with correct units. Getting comfortable with signed area and units now makes later topics like the Fundamental Theorem of Calculus and applications of integration much easier.

Key Takeaways

• The area between a rate of change graph and the x-axis gives the accumulated change over that interval.
• When the region is made of basic shapes (rectangles, triangles, trapezoids, semicircles), you can find accumulation exactly with geometry.
• Area below the x-axis counts as negative (signed area), so a negative rate means the quantity is decreasing.
• A positive rate over an interval gives positive accumulated change; a negative rate gives negative accumulated change.
• The units of accumulated change are the rate units multiplied by the units of the independent variable (for example, miles/hour times hours gives miles).
• Riemann sums let you estimate accumulation with rectangles when the region is not a clean geometric shape.

Change Over Time

Accumulation of change is the running total of how much a quantity changes. Suppose a car travels at a constant speed of 65 miles per hour for 3 hours. You can find the distance with:

$d=v\cdot t$

where d is distance, v is velocity, and t is time. Plugging in:

$d=\frac{65\text{ miles}}{\text{hour}}\cdot3\text{ hours}=\boxed{195 \text{ miles}}$

The unit hours in the denominator of one factor cancels with hours in the other, leaving miles. This is how units work for every accumulation problem: the result is the unit of the rate of change multiplied by the unit of the independent variable.

Graphing Change Over Time

You can show the relationship between speed, time, and distance on a graph where the x-axis is time and the y-axis is speed. The shaded area under the curve represents the total distance traveled.

A constant speed gives a rectangle, which is easy. But a real trip changes speed. Picture a car that leaves home, accelerates to 50 miles per hour, cruises on the highway for almost two hours, slows to 20 miles per hour, finishes the drive, then stops. How far did it travel?

This piecewise function describes the speed curve:

$$f(x) = \begin{cases} 5000x^2 & \text{if }0 \leq x \leq 0.1 \\ 50 & \text{if } 0.1\leq x \leq 2 \\ -5x^4+40x+50 & \text{if } 2\leq x \leq 2.205 \\ 20 & \text{if } 2.205\leq x \leq 2.75 \\ -80x+240 & \text{if } 2.75 \leq x \leq 3 \end{cases}$$

The area under this curve still gives distance, but it is not a simple rectangle. You can approximate it with rectangles or find it exactly with an integral.

Riemann Sums

A Riemann sum approximates area under a curve by drawing rectangles (or other easy shapes) and adding up their areas. Here is the process applied to the trip above.

Step 1: Choose How Many Rectangles

Decide how many rectangles to use. For this example, use six with equal widths. Divide the total base by the number of rectangles. The trip lasts 3 hours, so each base is 3 divided by 6, which is 0.5.

Step 2: Build the Rectangles

Place six rectangles of width 0.5 across the graph and read off a height for each one.

Step 3: Compute

List the base and height of each rectangle:

Add the areas:

$$(0.5\cdot0)+(0.5\cdot50)+(0.5\cdot50)+(0.5\cdot50)+(0.5\cdot50)+(0.5\cdot20)$$

Since every base is the same, factor it out:

$$0.5(0+50+50+50+50+20)=\boxed{110 \text{ miles}}$$

This approximation gives about 110 miles. It is not exact, since the rectangles include some overestimates and some underestimates. Exact methods come from integrals.

Preview of Integrals

Approximating area under the curve gives accumulation of change, but the integral finds the exact area. Integrals are also called antiderivatives because they undo differentiation. You will work with these in detail later in the unit.

Integration uses this notation:

$$\int_0^3 f(x) \ dx$$

This says to find the area under the curve over the interval [0,3]. This is a definite integral. For the trip example, the exact area works out to 117.44 miles, which is closer to the truth than the rough Riemann sum estimate.

How to Use This on the AP Calculus Exam

Problem Solving

• When you see a rate of change graph, the area between it and the x-axis is the accumulated change. Look for shapes you already know how to measure.
• Break a region into rectangles, triangles, trapezoids, or semicircles and use area formulas to get an exact value.
• Track signed area carefully. Area below the x-axis is negative and lowers the total accumulation.

Interpreting Answers

• Always attach units. Multiply the rate units by the independent variable units (for example, gallons/minute times minutes gives gallons).
• Connect your number to the situation. A velocity graph gives displacement, a flow rate gives total volume, and a growth rate gives change in population.

Common Trap

• Do not confuse the value of a rate at a point with the accumulated change over an interval. The height of the graph is the rate; the area is the accumulation.

Worked Example

Consider a graph of $f(x)$ made entirely of straight line segments, representing a vehicle's motion over time. What is the total distance toward its destination?

Because the graph is composed of straight lines, find the area of each piece geometrically.

• The first triangle has a base of 1 and a height of 10. Using $\frac{b\cdot h}{2}$, its area is 5.
• The next triangle has a base of 1.5 and a height of -10, giving an area of -7.5. This is signed area, so it is negative. In context, the car moves opposite to its destination.
• The next triangle has a base of 0.5 and a height of 10, for an area of 2.5.
• The large rectangle has a base of 3 and a height of 10, so $b\cdot h$ gives 30.
• The triangle on top of it has a base of 1 and a height of 20, for an area of 10.
• The last triangle has a base of 1 and a height of 30, for an area of 15.

Sum the areas:

$$5-7.5+2.5+30+10+15=55$$

Units are the rate (miles/hour) times the independent variable (hours), which gives miles. The car traveled a total of 55 miles toward its destination.

Practice Problems

Problem 1: A 50-gallon bathtub is being filled at a constant rate of 10 gal/minute. How long until the bathtub is full?

Problem 2: What is the total exact area under a curve made of three trapezoids, computed geometrically?

Problem 3: Estimate the area beneath the curve of $f(x)=-x^2+10x$ using a Riemann sum with 5 rectangles.

Solutions

Problem 1: You are given the rate of change and the final accumulation, and you need the time. From the first car example, accumulation equals rate times time: $a=r\cdot t$, where $a$ is accumulation, $r$ is the rate, and $t$ is time. Solve for time: $t=\frac{a}{r}$. So $t=50/10=\boxed{5\text{ minutes}}$.

Problem 2: The graph is made of three trapezoids. The area of a trapezoid is

$$\frac{a+b}{2}\cdot h$$

where $a$ and $b$ are the two base lengths and $h$ is the height. The trapezoids on the right and left are identical, with a height of 4 and base lengths of 4 and 6, so each has an area of 20. The trapezoid below the x-axis has a height of 4 and base lengths of 2 and 4, giving a signed area of -12. Summing: $20+20-12=28$ square units.

Problem 3: Draw 5 rectangles. The graph runs from 0 to 10, so each rectangle has a base of 2. Use the y-values for the heights:

$$2(0+16+24+24+16)=\boxed{160\ u^2}$$

Common Misconceptions

• Integration is not just "differentiation backwards." Even though an antiderivative undoes a derivative, choosing the right approach takes strategy. Reversing steps blindly leads to mistakes.
• The height of a rate graph is not the accumulated change. The height is the rate at that moment; the area under the graph is the accumulation over the interval.
• Area below the x-axis is negative. Forgetting signed area gives a total that is too large. A negative rate means the quantity is decreasing.
• Units are not optional. Accumulation units come from multiplying rate units by the independent variable units. Leaving them off or using the rate's units alone is incorrect.
• A Riemann sum is an estimate, not the exact area. With curved regions, rectangles over and undershoot the true value, so the sum is only an approximation unless the region is made of exact geometric shapes.

Related AP Calculus Guides

• Unit 6 Overview: Integration and Accumulation of Change
• 6.11 Integrating Using Integration by Parts
• 6.12 Integrating Using Linear Partial Fractions
• 6.5 Interpreting the Behavior of Accumulation Functions Involving Area
• 6.4 The Fundamental Theorem of Calculus and Accumulation Functions
• 6.13 Evaluating Improper Integrals