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CalculusBy Educator Today

C33 · Unit C8 Applications of integration

Average value and accumulated change

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 8.1, 8.3
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

The average value of a continuous function on [a,b] is (1/(b minus a))∫ₐᵇf(x)dx. It is the height of a rectangle with the same signed area, not the average of endpoint values except in special cases. If R is a rate of change, quantity Q(t) equals Q(a)+∫ₐᵗR(s)ds. Initial amount and accumulated change must be combined.

Worked example

A tank begins with 30 L and has net inflow R(t) equals 4 minus t L over min for 0 is less than or equal to t is less than or equal to 6. After six minutes the change is [4t minus t squared /2] sub 0 ⁶ equals 6 L, so it contains 36 L. The average net rate is 6 over 6 equals 1 L over min.

Explore the mathematics

Average value and accumulated change

Find a constant-height rectangle with the same accumulated area.

Setting 3 of 5

Scroll sideways to see the full diagram.

What constant-height rectangle has the same area as the curve?xy123Marked input 1.5; displayed curve height 2.25Average value of x² on [0,2]

Predict before continuing

What constant-height rectangle has the same area as the curve?

Read the complete explanation
  1. What constant-height rectangle has the same area as the curve?
  2. Accumulated area is ∫₀²x² dx=8/3.
  3. Divide by interval length 2−0.
  4. The average value is 4/3, not the value at the midpoint, which is 1.
See the idea: Average value and accumulated change
See the idea: Average value and accumulated changeThe net rate R(t)=4−t is positive until t=4 and negative afterwards. Signed area on [0,6] is 8−2=6 L. Adding the initial 30 L gives 36 L; the average net rate is 1 L/min.xy0-31.5-1314.5365

The net rate R(t)=4−t is positive until t=4 and negative afterwards. Signed area on [0,6] is 8−2=6 L. Adding the initial 30 L gives 36 L; the average net rate is 1 L/min.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

An integral of a rate gives a change, not automatically the final amount. Average value divides by interval length.

Investigate and explain

Compare average height and average rate on the same graph. Explain why their formulas can look alike but their units differ.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.