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C21 · Unit C6 Integration and accumulation

Accumulation, Riemann sums and sigma notation

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 6.1–6.3
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

A rate multiplied by a small time width approximates change. A Riemann sum sum f(x sub i *) delta x sub i adds these pieces. As the largest width tends to zero, the sum defines the definite integral for an integrable function. Signed heights count downward accumulation negatively. Left, right and midpoint rules use different sample positions, and unequal-width tables require a separate width for every interval.

Worked example

For f(t) equals t squared on [0,2] with two equal intervals, the left sum is 0 squared times 1+1 squared times 1 equals 1, right sum 1 squared +2 squared equals 5, and midpoint sum 0.5 squared +1.5 squared equals 2.5. The exact integral is 8 over 3; all three are estimates.

Explore the mathematics

Accumulation, Riemann sums and sigma notation

Change the partition and compare an area estimate with its exact integral.

Setting 3 of 5

Scroll sideways to see the full diagram.

Predict whether this area estimate is above or below the exact value.xy12y=x², [0,2]; 6 rectangles

Predict before continuing

Predict whether this area estimate is above or below the exact value.

Read the complete explanation
  1. Predict whether this area estimate is above or below the exact value.
  2. Each piece has width 0.3333. Its area depends on the chosen height.
  3. Add the piece areas: 3.3704. The exact integral is 8/3≈2.6667.
  4. Increase the number of pieces. The estimate approaches the integral; a finite sum generally has approximation error.
Accumulation, Riemann sums and sigma notation · visual worked method
Accumulation, Riemann sums and sigma notation · visual worked methodFor f(t)=t² on [0,2] with two equal intervals, the left sum is 0²·1+1²·1=1, right sum 1²+2²=5, and midpoint sum 0.5²+1.5²=2.5. The exact integral is 8/3; all three are estimates.Worked example · 1/2For f(t)=t² on [0,2] with twoequal intervals, the left sum is0²·1+1²·1=1, right sum 1²+2²=5,and midpoint sumWorked example · 2/20.5²+1.5²=2.5. The exact integralis 8/3; all three are estimates.

For f(t)=t² on [0,2] with two equal intervals, the left sum is 0²·1+1²·1=1, right sum 1²+2²=5, and midpoint sum 0.5²+1.5²=2.5. The exact integral is 8/3; all three are estimates.

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

A common misconception

Do not multiply by a common width if the table spacing is unequal. An integral of a rate has quantity units.

Investigate and explain

Draw unequal-width rectangles from the table. Explain why an unweighted average of rates is not the integral.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.