6.1 - Exploring Accumulations of Change

Understanding “Area Under the Curve”

Important definition: “area under the curve” means the area between the function and the x-axis, not just the area below the function.

The region is bounded by: 1. The curve 2. The x-axis

The phrase “under the curve” is a little misleading. It doesn’t mean “below” in a spatial sense. It means “between the curve and the x-axis.”

Accumulation from Rates of Change

Here’s the fundamental idea: if a function represents a rate of change, then the area under the curve gives the accumulation of change.

\boxed{\text{Rate of change function} \rightarrow \text{Area under curve} = \text{Total accumulation}}

Simple example: constant speed

Drive at a constant 60 mph for 4 hours.

Traditional calculation:

\text{Distance} = 60 \frac{\text{miles}}{\text{hour}} \times 4 \text{ hours} = 240 \text{ miles}

Area interpretation:

\text{Area} = \text{width} \times \text{height} = 60 \frac{\text{miles}}{\text{hour}} \times 4 \text{ hours} = 240 \text{ miles}

The area is the total distance traveled, the accumulation of the rate of change.

Units for Area Under a Curve

\boxed{\text{Area units} = (\text{y-axis units}) \times (\text{x-axis units})}

Or equivalently, (dependent variable units) × (independent variable units).

Speed and time:

\frac{\text{miles}}{\text{hour}} \times \text{hours} = \text{miles}

The hours cancel, leaving miles (distance).

Flow rate and time:

\frac{\text{gallons}}{\text{minute}} \times \text{minutes} = \text{gallons}

The minutes cancel, leaving gallons (total volume).

Example 1: Water Leaking from a Tank

Water leaks from a tank at varying rates. The graph shows the leak rate in gallons per minute over 9 minutes.

Find the total water leaked in 9 minutes. Just find the area of each geometric shape.

Rectangle (0 to 3 minutes):

\text{Area} = 3 \times 4 = 12

Semicircle (3 to 7 minutes), radius 2:

\text{Area} = \frac{\pi r^2}{2} = \frac{\pi (2)^2}{2} = 2\pi

Triangle (7 to 9 minutes), base 2 and height 3:

\text{Area} = \frac{1}{2} \times 2 \times 3 = 3

Total:

12 + 2\pi + 3 = 15 + 2\pi \text{ gallons}

That’s the exact answer. Approximately,

15 + 2(3.14159) \approx 21.28

gallons.

Interpreting the answer in context: on the AP exam, a complete interpretation names the quantity, gives the amount with units, and says the time interval. Here: a total of

15 + 2\pi \approx 21.28

gallons of water leaked out of the tank during the first 9 minutes.

Example 2: Particle Motion with Direction Changes

A particle moves along the x-axis with the velocity shown below, in cm/s, with time in seconds.

Part A: how far is the particle from its starting position after 10 seconds?

Find each area, keeping track of signs.

Region 1 (0 to 2 s, above axis): triangle, base 2, height 10

\text{Area}_1 = \frac{1}{2}(2)(10) = 10 \quad \text{(positive)}

Region 2 (2 to 5 s, below axis): triangle, base 3, height 15

\text{Area}_2 = \frac{1}{2}(3)(15) = 22.5 \quad \text{(negative)}

Region 3 (5 to 7 s, below axis): rectangle, base 2, height 15

\text{Area}_3 = 2 \times 15 = 30 \quad \text{(negative)}

Region 4 (7 to 8 s, below axis): triangle, base 1, height 15

\text{Area}_4 = \frac{1}{2}(1)(15) = 7.5 \quad \text{(negative)}

Region 5 (8 to 10 s, above axis): triangle, base 2, height 10

\text{Area}_5 = 10 \quad \text{(positive)}

Net displacement:

10 - 22.5 - 30 - 7.5 + 10 = 20 - 60 = -40

The particle ends up 40 cm to the left of its starting position. The negative sign is direction (left along the x-axis).

Part B: how far did the particle travel between 2 and 8 seconds?

Between

t = 2

and

t = 8

, the particle is moving in the negative direction the whole time (velocity is below the axis).

22.5 + 30 + 7.5 = 60

The particle traveled 60 cm to the left.

Positive vs. Negative Area

Sign convention: - Area above the horizontal axis: positive - Area below the horizontal axis: negative

For motion problems: - Positive area means movement in the positive direction (right, forward, up) - Negative area means movement in the negative direction (left, backward, down)

Net displacement: add positive areas, subtract negative ones.

Total distance traveled: add the absolute values of all areas, ignoring signs.

Strategy for Geometric Shapes

Shape	Area Formula
Rectangle	$A = \text{base} \times \text{height}$
Triangle	$A = \frac{1}{2} \times \text{base} \times \text{height}$
Semicircle	$A = \frac{\pi r^2}{2}$
Trapezoid	$A = \frac{1}{2}(b_1 + b_2) \times h$

Tip: break complicated regions into simple shapes, find each area, and add them up.

Key Takeaways

“Area under curve” means the area between the curve and the x-axis, not necessarily below the curve

Rate of change function → area gives the total accumulation of change

Units for area: (y-axis units) × (x-axis units), with the appropriate cancellation

Positive area: where the curve is above the horizontal axis

Negative area: where the curve is below the horizontal axis

Net change: positive areas minus negative areas

Total amount: sum of the absolute values of all areas

Geometric shapes: use standard area formulas for rectangles, triangles, circles

Direction matters: the sign of the area tells you the direction of change in motion problems

Always include proper units in your final answer

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