Riemann Sum

Estimating Area with Rectangular Strips

Imagine a plot of land with one irregular curved edge. To estimate its area, we can divide the plot into rectangular strips of equal width. We then calculate the area of each strip and add the results.

A Riemann sum divides an interval into subintervals, chooses one sample point in each, and adds the signed rectangle contributions. For a Riemann-integrable function, these sums approach the definite integral as the widest subinterval approaches zero. When the function is nonnegative, the signed accumulation equals the area under the curve. With equal-width partitions, increasing the number of rectangles makes each one narrower, although one additional rectangle does not always move the sum closer to the definite integral.

Interval Partition and Width

Four quantities define an equal-width Riemann sum:

The sample point can be the left endpoint, right endpoint, midpoint, or another point inside the subinterval.

The Riemann Sum Formula

If we combine all these components, we get the general formula for a Riemann Sum:

The sigma notation ($\sum$) simply means "sum up all the areas of the rectangles," where the area of each rectangle is its height ($f(x_i^*)$) times its width ($\Delta x$).

Left Endpoint Sum for a Linear Function

Problem: Determine the Riemann Sum for the function $f(x) = x$ on the interval $[0, 7]$ by dividing it into $7$ subintervals of equal length and using the left endpoint as the sample point.

Solution:

1. Identify Components:

   • Function: $f(x) = x$
   • Interval: $a=0, b=7$
   • Number of partitions: $n=7$

2. Calculate Subinterval Width ($\Delta x$):

   $$\Delta x = \frac{7 - 0}{7} = 1$$

   Each rectangle will have a width of $1$.

3. Determine the Sample Points (Left Endpoint):

   Our subintervals are:

   $$\begin{aligned} &[0,1], [1,2], [2,3], [3,4], [4,5], \\ &[5,6], [6,7] \end{aligned}$$

   For the left-endpoint method, we take the value of $x$ from the left side of each subinterval as the sample point:

   $$\begin{aligned} &x_1=0, x_2=1, x_3=2, x_4=3, \\ &x_5=4, x_6=5, x_7=6 \end{aligned}$$

4. Calculate the Height of Each Rectangle:

   The height of each rectangle is determined by the function's value $f(x) = x$ at the chosen sample points:

   $$f(x_1) = f(0) = 0$$

   $$f(x_2) = f(1) = 1$$

   $$f(x_3) = f(2) = 2$$

   $$f(x_4) = f(3) = 3$$

   $$f(x_5) = f(4) = 4$$

   $$f(x_6) = f(5) = 5$$

   $$f(x_7) = f(6) = 6$$

5. Calculate the Riemann Sum:

   Now we can calculate the total area by adding the areas of all rectangles. The area of each rectangle is its height times its width:

   $$R_7 = \sum_{i=1}^{7} f(x_i^*) \Delta x$$

   Denote the sum of the seven sampled values by $S$:

   $$\begin{aligned} S &=f(0)+f(1)+f(2)+f(3) \\ &\quad{}+f(4)+f(5)+f(6) \end{aligned}$$

   $$=S\cdot1$$

   $$= (0 + 1 + 2 + 3 + 4 + 5 + 6) \cdot 1$$

   $$= 21$$

   Thus, the estimated area under the curve $f(x)=x$ from $0$ to $7$ is $21$.

On $[0,7]$, the function $f(x)=x$ is nonnegative and strictly increasing. Therefore, rectangles built from left endpoints lie below the graph and underestimate the actual area, while rectangles built from right endpoints overestimate it.

Exercises

Each problem asks for an area approximated by rectangles, so fix the number of subintervals and the sample point in each one before you add the terms.

1. Calculate the Riemann Sum for the function $f(x) = x^2 + 1$ on the interval $[0, 4]$ using $4$ subintervals of equal width and the right endpoint as the sample point.

Worked Solution

1. We will calculate the Riemann Sum for $f(x) = x^2+1$ on $[0,4]$ with $n=4$.

   Riemann Sum Exercise Visualization

   Graph of the function $f(x)=x^2+1$ with $4$ rectangular partitions using the right endpoint.

   GridAutomatic rotation

   Step 1: Determine the main components.

   • Function: $f(x) = x^2 + 1$
   • Interval: $a=0, b=4$
   • Number of partitions: $n=4$
   • Sample point: Right endpoint

   Step 2: Calculate the subinterval width.

   $$\Delta x = \frac{b-a}{n} = \frac{4-0}{4} = 1$$

   Step 3: Determine the right endpoint sample points.

   The subintervals are:

   $$[0,1], [1,2], [2,3], [3,4]$$

   For the right-endpoint method, we take the value of $x$ from the right side of each subinterval:

   $$x_1=1, x_2=2, x_3=3, x_4=4$$

   Step 4: Calculate the height of each rectangle.

   The height of each rectangle is determined by the function's value $f(x) = x^2 + 1$ at the right endpoint sample points:

   $$f(x_1) = f(1) = 1^2 + 1 = 2$$

   $$f(x_2) = f(2) = 2^2 + 1 = 5$$

   $$f(x_3) = f(3) = 3^2 + 1 = 10$$

   $$f(x_4) = f(4) = 4^2 + 1 = 17$$

   Step 5: Calculate the total Riemann Sum.

   Now we sum the areas of all rectangles ($\text{height} \times \text{width}$):

   $$R_4 = \sum_{i=1}^{4} f(x_i^*) \Delta x$$

   $$= [f(1) + f(2) + f(3) + f(4)] \cdot 1$$

   $$= (2 + 5 + 10 + 17) \cdot 1$$

   $$= 34$$

   Thus, the Riemann Sum for the function is $34$.