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CalculusBy Educator Today

C50 · Unit C6 Integration and accumulation

Turning a sum into an integral

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

Progress on this lesson counts in every pathway listed.

Understand the idea

A Riemann sum has three pieces: an interval, a width and a sample point. On [a,b], equal width is delta x equals (b minus a)/n and right endpoints are x sub i equals a+i delta x. Recognize sum f(x sub i) delta x before taking n tends to ∞. A factor involving n may be the width rather than part of the function.

Worked example

lim sum from i equals 1 to n of (2 over n)(1+2i over n) squared is ∫ sub 1 cubed x squared dx equals 26 over 3. Here delta x equals 2 over n and x sub i equals 1+2i over n. The interval starts at 1 and has length 2; it is not [0,2].

Explore the mathematics

Turning a sum into an integral

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.
Turning a sum into an integral · visual worked method
Turning a sum into an integral · visual worked methodlim Σ from i=1 to n of (2/n)(1+2i/n)² is ∫₁³x²dx=26/3. Here Δx=2/n and xᵢ=1+2i/n. The interval starts at 1 and has length 2; it is not [0,2].Worked example · 1/2lim Σ from i=1 to n of(2/n)(1+2i/n)² is ∫₁³x²dx=26/3.Here Δx=2/n and xᵢ=1+2i/n. Theinterval starts at 1 and haslength 2;Worked example · 2/2it is not [0,2].

lim Σ from i=1 to n of (2/n)(1+2i/n)² is ∫₁³x²dx=26/3. Here Δx=2/n and xᵢ=1+2i/n. The interval starts at 1 and has length 2; it is not [0,2].

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

A common misconception

Inferring limits from the coefficient alone loses the starting input. Reconstruct both a and b.

Investigate and explain

Color-code the sample point and width in three sum expressions before converting them to integrals.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.