Search topics

Search

Approximate area by an endpoint or trapezium rule

Start a set of 8
Step 2 of 7

Use the rectangle rules

Partition the interval into equal subintervals and use one endpoint from each subinterval. Multiply the selected heights by the common width and add the rectangle areas

\[L=w\sum_{i=0}^{n-1}f(a+iw),\quad R=w\sum_{i=1}^{n}f(a+iw)\]
  1. 1Find the common width Divide the interval length by the number of subintervals \(\displaystyle w=\frac{b-a}{n}\)
  2. 2Choose the stated endpoint A left rule uses the first endpoint of each subinterval, while a right rule uses the second endpoint \(\displaystyle L=w\left(f(a)+f(a+w)+\cdots+f(b-w)\right)\)
  3. 3Multiply each height by the width The area of each rectangle is its height multiplied by the common width \(\displaystyle A_i=w\times f(x_i)\)
  4. 4Compare with the curve direction For an increasing function, left rectangles underestimate and right rectangles overestimate the area \(\displaystyle f\text{ increasing}\quad\Rightarrow\quad L<\int_a^b f(x)\,dx<R\)
Year 12, VCE Units 3 & 4, Chapter 5, 1 difficulty level, Calculator allowedBuild a worksheet

Use the rectangle rules

Partition the interval into equal subintervals and use one endpoint from each subinterval. Multiply the selected heights by the common width and add the rectangle areas

\[L=w\sum_{i=0}^{n-1}f(a+iw),\quad R=w\sum_{i=1}^{n}f(a+iw)\]
  1. Find the common width Divide the interval length by the number of subintervals \(\displaystyle w=\frac{b-a}{n}\)
  2. Choose the stated endpoint A left rule uses the first endpoint of each subinterval, while a right rule uses the second endpoint \(\displaystyle L=w\left(f(a)+f(a+w)+\cdots+f(b-w)\right)\)
  3. Multiply each height by the width The area of each rectangle is its height multiplied by the common width \(\displaystyle A_i=w\times f(x_i)\)
  4. Compare with the curve direction For an increasing function, left rectangles underestimate and right rectangles overestimate the area \(\displaystyle f\text{ increasing}\quad\Rightarrow\quad L<\int_a^b f(x)\,dx<R\)

Worked example

Use the left-endpoint rectangle rule with three equal subintervals to approximate the area under the curve

\[f(x)=3x+2,\quad 1\le x\le4,\quad w=1\]
  1. \(\displaystyle x=1,2,3,\quad w=\frac{4-1}{3}=1\)The left endpoints are 1, 2 and 3, and the common width is 1.
  2. \(\displaystyle f(1)=5,\quad f(2)=8,\quad f(3)=11\)Evaluate the function at each left endpoint
  3. \(\displaystyle L=1\left(5+8+11\right)=24\)Multiply the sum of the selected heights by the common width
  4. \(\displaystyle A\approx24\text{ square units}\)The left-endpoint rectangle approximation is 24 square units

Common mistakes

Using the wrong endpoint
✕\(\displaystyle L=1\left(f(1)+f(2)+f(3)\right)\)✓\(\displaystyle L=1\left(f(0)+f(1)+f(2)\right)\)
For three unit-width subintervals from 0 to 3, the left endpoints are 0, 1 and 2, not 1, 2 and 3
Omitting the common width
✕\(\displaystyle L=4+5+6=15\)✓\(\displaystyle L=\frac{1}{2}\left(4+5+6\right)=\frac{15}{2}=7.5\)
Each rectangle area is its height multiplied by the common width, so the width cannot be omitted
Using one extra rectangle
✕\(\displaystyle L=1\left(f(0)+f(1)+f(2)+f(3)\right)\)✓\(\displaystyle L=1\left(f(0)+f(1)+f(2)\right)\)
Three subintervals need three selected heights, so a left rule does not include the far endpoint

Difficulty levels

introUse a left or right rule for a polynomial with unit width and exact heights

Questions about this topic

What is the difference between a left rule and a right rule?+

A left rule uses the left endpoint of each subinterval for its height. A right rule uses the right endpoint

Why is the common width multiplied by the height sum?+

Each rectangle has area equal to width times height, so the common width scales the sum of all selected heights

When does a rectangle rule underestimate the area?+

For an increasing function, the left rule underestimates and the right rule overestimates. The relationship reverses for a decreasing function

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Specialist Mathematics, chapter 5: Integral calculus

In this chapter

Integrate inverse circular kernels and change variablesAnti-differentiate reciprocal and linear-reciprocal functionsIntegration by recognition with logarithmic componentsAnti-differentiate by recognition from a supplied derivativeIntegrate using partial fractions, identities and partsDecompose rational expressions into partial fractionsCompound, double-angle and reciprocal identitiesAnti-differentiate exponential and circular functionsFind exact areas and assess numerical integration resultsFind the area between two curvesApproximate area by an endpoint or trapezium ruleFind areas involving reciprocal curvesAudit numerical integration loops and error enclosuresSet up volumes, parametric lengths and surfaces of revolution