Approximate area by an endpoint or trapezium rule

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

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

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

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