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CalculusBy Educator Today

C22 · Unit C6 Integration and accumulation

Trapezoids and integral properties

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 6.2, 6.6
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

The trapezoidal estimate joins endpoint heights linearly on each subinterval: sum one half (f sub i +f sub i plus 1) delta x sub i. It is exact for linear functions and overestimates a concave-up function because secants lie above the graph. Integral properties include additivity over adjacent intervals, linearity in the integrand, and sign reversal when bounds swap.

Worked example

For t equals 0,2,5 and r equals 3,7,4, the trapezoidal estimate is one half (3+7) times 2+ one half (7+4) times 3 equals 10+16.5 equals 26.5. Unequal widths 2 and 3 are essential. This is net accumulation if the data are signed rates.

Explore the mathematics

Trapezoids and integral properties

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 trapezoids

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: 2.7037. 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.
Trapezoids and integral properties · visual worked method
Trapezoids and integral properties · visual worked methodFor t=0,2,5 and r=3,7,4, the trapezoidal estimate is ½(3+7)·2+½(7+4)·3=10+16.5=26.5. Unequal widths 2 and 3 are essential. This is net accumulation if the data are signed rates.Worked example · 1/2For t=0,2,5 and r=3,7,4, thetrapezoidal estimate is½(3+7)·2+½(7+4)·3=10+16.5=26.5.Unequal widths 2 and 3 areessential.Worked example · 2/2This is net accumulation if thedata are signed rates.

For t=0,2,5 and r=3,7,4, the trapezoidal estimate is ½(3+7)·2+½(7+4)·3=10+16.5=26.5. Unequal widths 2 and 3 are essential. This is net accumulation if the data are signed rates.

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

A common misconception

Trapezoid accuracy cannot be inferred solely from monotonicity; curvature controls a consistent over over understatement.

Investigate and explain

Compare left, right and trapezoidal estimates for one increasing concave-down function. Explain the source of each inequality.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.