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Wize University Calculus 1 Textbook > Integrals

Approximating Areas & Riemann Sums

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Approximating Areas & Riemann Sums

Areas of planar regions can be approximated (estimated) using finite sums and this process is easier when the region is bounded by the graph of a function.

To approximate the area under the curve f(x)f\left(x\right) on the interval [a, b]\left[a,\ b\right], we divide the area into rectangles with equal widths Δx\Delta x. Each rectangle will have height f(x)f\left(x\right). Then, the approximate area is the sum of the areas of all of these rectangles.


Wize Tip
Depending on how we draw these rectangles, we will get slightly different area approximations.

Common Approximation Rules/Methods

  • Right Endpoint Rule: The area is estimated using rectangles whose right endpoint touches the graph of the function.
  • Left Endpoint Rule: The area is estimated using rectangles whose left endpoint touches the graph of the function.
  • Midpoint Rule: The area is estimated using rectangles whose midpoint touches the graph of the function.
  • Lower Sum: The area is estimated using rectangles lying inside the region, with one point touching the graph.

The sum of the areas of these rectangles is called a Riemann Sum.
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Riemann Sum

Let f(x)f(x) be a function and xix_i a set of points in its domain. We can estimate the area under the graph of f(x)f(x) on the interval [a,b][a,b] using the Riemann Sum

∑i=1nf(xi)Δx\boxed{\sum_{i=1}^{n}f(x_i)\Delta x }
Where Δx\Delta x is a constant of the form

Δx=b−an\displaystyle \boxed{\Delta x=\frac{b-a}{n}}

  • xi=a+iΔxx_i=a+i\Delta x is a right hand rule
  • xi=a+(i−1)Δxx_i=a+(i-1)\Delta x is a left hand rule
  • xi=a+(i−1/2)Δxx_i=a+(i-1/2)\Delta x is a midpoint rule
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Example: Riemann Sums

Estimate the area under the curve and over [0,1][0,1] of the function y=1−x2y=1-x^2 using 4 rectangles.
First divide [0,1][0,1] into 4 equal sub-intervals:
Δx=1−04=14 \Delta x = \dfrac{1-0}{4} = \dfrac{1}{4}
So, the sub-intervals are
[0,14],[14,12],[12,34],[34,1]\left[0,\frac{1}{4}\right],\left[\frac{1}{4},\frac{1}{2}\right],\left[\frac{1}{2},\frac{3}{4}\right],\left[\frac{3}{4},1\right]

If we like to use the left endpoint method, we should take x0=0x_0=0, x1=14x_1=\frac{1}{4}, x2=12x_2=\frac{1}{2}, x3=34x_3=\frac{3}{4}
Consider the table


Now the approximate area is
A≈∑i=14f(xi−1)Δx=[f(x0)+f(x1)+f(x2)+f(x3)]Δx=[1+1516+34+716](14)=0.78125 \begin{aligned} A \approx \sum_{i=1}^4f(x_{i-1})\Delta x=& [f(x_0)+f(x_1)+f(x_2)+f(x_3)]\Delta x\\ =& \left[1+\frac{15}{16}+\frac{3}{4}+\frac{7}{16}\right]\left(\frac{1}{4}\right) = 0.78125 \end{aligned}

*If you use another rule/method such as the midpoint rule or right endpoint rule to approximate the area, you'll get a slightly different approximation
Use the left-endpoint Riemann sum, with n=4n=4, to estimate the area under the curve f(x)=ex−xf\left(x\right)=e^x-x on the interval [1, 13]\left[1,\ 13\right].
Extra Practice
Practice: Definite Integrals and Area (~F2018 Final Q31)
Practice: Definite Integrals and Area
What is the integral that equals lim⁡n→∞Σi=1n(1n)[3(1+in)3+3]\displaystyle\lim_{n\rightarrow\infty}\Sigma_{i=1}^n\left(\frac{1}{n}\right)\left[3\left(1+\frac{i}{n}\right)^3+3\right]?
Practice: Area Using Sigma Notation (~F2014 Final Q17)
Practice Question: Area Using Sigma Notation
Determine the xx boundaries of the region whose area is lim⁡n→∞ Σi=1n 5n [8−(4+5in)2]\lim_{n\rightarrow\infty}\ \Sigma_{i=1}^n\ \frac{5}{n}\ \left[8-\left(4+\frac{5i}{n}\right)^2\right].
Practice: Area using Sigma Notation (~F2018 Final Q31)
Practice: Area using Sigma Notation
The area bounded by the function f(x)f\left(x\right) and the xx-axis in the interval [a, b]\left[a,\ b\right] is represented by lim⁡n→∞ Σi=1n(3n)[cos⁡(2+3in)(2+3in)]\lim_{n\rightarrow\infty}\ \Sigma_{i=1}^n\left(\frac{3}{n}\right)\left[\frac{\cos\left(2+\frac{3i}{n}\right)}{\left(2+\frac{3i}{n}\right)}\right] , find the function f(x)f\left(x\right) and the interval [a, b]\left[a,\ b\right].
Practice: Approximating Rectangles
Q.\textbf{Q.} Estimate the area under the curve y=1−x2y=1-x^2 on the interval [0,1][0,1] using 4 rectangles and left end-points.
Identify Integral from Definition: Riemann Sums
The given expression represents the limit of the sum of areas of nn rectangles approximating the area of a certain region in the plane. Identify f(x), Δxf\left(x\right),\ \Delta x and the interval [a,b]\left[a,b\right] which correspond to this expression, and write it as a definite integral: lim⁡n→∞∑i=1nsin⁡(πin)⋅πn\displaystyle \lim_{n\to\infty}\sum_{i=1}^n\sin\left(\frac{\pi i}{n}\right)\cdotp\frac{\pi}{n}
Approximating Rectangles: Riemann Sums
Approximate the area under the curve of the function f(x)=1xf(x)=\dfrac{1}{x} from x=2x=2 to x=5x=5 .
Approximating Rectangles: Riemann Sums
Approximate the area under the curve of the function f(x)=1xf(x)=\dfrac{1}{x} from x=2x=2 to x=5x=5 .
Approximating Areas: Definite Integrals
The limit lim⁡n→∞∑i=1n1n1−(−1+in)2\displaystyle\lim_{n\to\infty}\sum_{i=1}^n\frac{1}{n}\sqrt{1-\left(-1+\frac{i}{n}\right)^2} describes which of the following?
Riemann Sums
Evaluate lim⁡n→∞ ∑i=1n[5n[(1+5in)2−3]]\displaystyle \lim_{n\to\infty}\ \sum_{i=1}^n\left[\frac{5}{n}\left[\left(1+\frac{5i}{n}\right)^2-3\right]\right]
Evaluate Σi=110 [2(i−3i2)−1]\Sigma_{i=1}^{10}\ \left[2\left(i-3i^2\right)-1\right].
Evaluate Σi=110 [2(i−3i2)−1]\Sigma_{i=1}^{10}\ \left[2\left(i-3i^2\right)-1\right].
Find the bounds of the region whose area is
lim⁡n→∞Σi=1n(4n)((−2+4in)−3(−2+4in)4)\lim_{n\rightarrow\infty}\Sigma_{i=1}^n\left(\frac{4}{n}\right)\left(\left(-2+\frac{4i}{n}\right)-3\left(-2+\frac{4i}{n}\right)^4\right)
Evaluate lim⁡x→∞Σi=1n 1n⋅[e(−3+in)]\lim_{x\rightarrow\infty}\Sigma_{i=1}^n\ \frac{1}{n}\cdot\left[e^{\left(-3+\frac{i}{n}\right)}\right].
Riemann Sums: Approximating Areas
Approximate the area under the curve of
f(x)=2−cos⁡2 (πx)f\left(x\right)=2-\cos^{2\ }\left(\pi x\right) on the interval [0,8]\left[0,8\right] using a left hand Riemann Sum and 4 sub intervals.
Approximate the area under the curve of
f(x)=2−cos⁡2 (πx)f\left(x\right)=2-\cos^{2\ }\left(\pi x\right) on the interval [0,8]\left[0,8\right] using a left hand Riemann Sum and 4 sub intervals.
Approximating Areas
Using 3 rectangles (partition of size 3) and right end points, the area under f(x)=9−xf(x)=\sqrt{9-x} on the interval [0, 9] is estimated to be
Approximating Area and Riemann Sums
Use 4 left rectangles in a Riemann sum to approximate the area under y=x2y=x^2 on [0,2][0,2].
Riemann Sums: Approximating Areas
The area bounded by the function f(x)f\left(x\right) and the xx-axis in the interval [a,b]\left[a,b\right] is given by lim⁡n→∞∑i=1n4in2[ein]\displaystyle \lim_{n\to\infty}\sum_{i=1}^n\frac{4i}{n^2}\left[e^{\frac{i}{n}}\right].
Determine f(x)f(x), aa, and bb.
Approximating area under curve
Practice: Area using Riemann Sums
The area bounded by the function f(x)f\left(x\right) and the xx-axis in the interval [a, b]\left[a,\ b\right] is represented by lim⁡n→∞ ∑i=1n(3n)[cos⁡(2+3in)(2+3in)]\displaystyle \lim_{n\rightarrow\infty}\ \sum_{i=1}^n\left(\frac{3}{n}\right)\left[\frac{\cos\left(2+\frac{3i}{n}\right)}{\left(2+\frac{3i}{n}\right)}\right].
Find the function f(x)f\left(x\right) and the interval [a, b]\left[a,\ b\right].
Practice: Estimating Area Using Riemann Sum
Given the following table of values for the function f(x)f\left(x\right), find the left-endpoint Riemann sum L3L_3 and right-endpoint Riemann sum R3R_3 to estimate the area under the graph of f(x)f\left(x\right) on the interval [0, 6]\left[0,\ 6\right].
Identify Integral from Definition: Riemann Sums
The given expression represents the limit of the sum of areas of nn rectangles approximating the area of a certain region in the plane. Identify f(x), Δxf\left(x\right),\ \Delta x and the interval [a,b]\left[a,b\right] which correspond to this expression, and write it as a definite integral: lim⁡n→∞∑i=1nsin⁡(πin)⋅πn\displaystyle \lim_{n\to\infty}\sum_{i=1}^n\sin\left(\frac{\pi i}{n}\right)\cdotp\frac{\pi}{n}
Approximating Rectangles: Riemann Sums
Approximate the area under the curve of the function f(x)=1xf(x)=\dfrac{1}{x} from x=2x=2 to x=5x=5 .
Practice: Approximating Rectangles
Q.\textbf{Q.} Estimate the area under the curve y=1−x2y=1-x^2 on the interval [0,1][0,1] using 4 rectangles and left end-points.
Evaluate lim⁡x→∞Σi=1n 1n⋅[e(−3+in)]\lim_{x\rightarrow\infty}\Sigma_{i=1}^n\ \frac{1}{n}\cdot\left[e^{\left(-3+\frac{i}{n}\right)}\right].
Evaluate Σi=110 [2(i−3i2)−1]\Sigma_{i=1}^{10}\ \left[2\left(i-3i^2\right)-1\right].
Find the bounds of the region whose area is
lim⁡n→∞Σi=1n(4n)((−2+4in)−3(−2+4in)4)\lim_{n\rightarrow\infty}\Sigma_{i=1}^n\left(\frac{4}{n}\right)\left(\left(-2+\frac{4i}{n}\right)-3\left(-2+\frac{4i}{n}\right)^4\right)
Riemann Sums: Approximating Areas
Use the left-endpoint Riemann sum, with n=4n=4, to estimate the area under the curve f(x)=ex−xf\left(x\right)=e^x-x on the interval [1, 13]\left[1,\ 13\right].
Riemann Sums: Approximating Areas
Which of the following integrals can be used to represent the limit lim⁡n→∞ Σi=1n[3n(3in−1)sin⁡(3in+2)]\lim_{n\to\infty}\ \Sigma_{i=1}^n\left[\frac{3}{n}\left(\frac{3i}{n}-1\right)\sin\left(\frac{3i}{n}+2\right)\right]?
Riemann Sums: Approximation of areas under curves
If Δx=3n\Delta x=\frac{3}{n} andxi=iΔxx_i=i\Delta x, then what is the integral that represents lim⁡n→∞Σi=1nxi(1+xi)2Δx\lim_{n\rightarrow\infty}\Sigma_{i=1}^nx_i\left(1+x_i\right)^2\Delta x?
Practice: Definite Integrals and Area (~F2018 Final Q31)
Practice: Definite Integrals and Area
What is the integral that equals lim⁡n→∞Σi=1n(1n)[3(1+in)3+3]\displaystyle\lim_{n\rightarrow\infty}\Sigma_{i=1}^n\left(\frac{1}{n}\right)\left[3\left(1+\frac{i}{n}\right)^3+3\right]?
Practice: Area Using Sigma Notation (~F2014 Final Q17)
Practice Question: Area Using Sigma Notation
Determine the xx boundaries of the region whose area is lim⁡n→∞ Σi=1n 5n [8−(4+5in)2]\lim_{n\rightarrow\infty}\ \Sigma_{i=1}^n\ \frac{5}{n}\ \left[8-\left(4+\frac{5i}{n}\right)^2\right].
Practice: Area using Sigma Notation (~F2018 Final Q31)
Practice: Area using Sigma Notation
The area bounded by the function f(x)f\left(x\right) and the xx-axis in the interval [a, b]\left[a,\ b\right] is represented by lim⁡n→∞ Σi=1n(3n)[cos⁡(2+3in)(2+3in)]\lim_{n\rightarrow\infty}\ \Sigma_{i=1}^n\left(\frac{3}{n}\right)\left[\frac{\cos\left(2+\frac{3i}{n}\right)}{\left(2+\frac{3i}{n}\right)}\right] , find the function f(x)f\left(x\right) and the interval [a, b]\left[a,\ b\right].
What is the integral that equals lim⁡n→∞Σi=1n(1n)[3(1+in)3+3]\lim_{n\rightarrow\infty}\Sigma_{i=1}^n\left(\frac{1}{n}\right)\left[3\left(1+\frac{i}{n}\right)^3+3\right]
What is the integral that equals lim⁡n→∞Σi=1n(1n)[3(1+in)3+3]\lim_{n\rightarrow\infty}\Sigma_{i=1}^n\left(\frac{1}{n}\right)\left[3\left(1+\frac{i}{n}\right)^3+3\right]