Riemann Sums and Area Approximation

Finding the exact area under a curve can be tricky. Riemann sums provide a practical way to approximate this area by breaking the region down into simpler geometric shapes, like rectangles or trapezoids.

What is a Riemann Sum?

A Riemann sum approximates the area under a curve $y = f(x)$ on an interval $[a, b]$ by dividing the interval into $n$ smaller subintervals and building a shape (usually a rectangle) on each one.

If we divide the interval into $n$ subintervals of equal width, the width of each subinterval is: $\Delta x = \frac{b - a}{n}$

Types of Rectangular Approximations

Depending on where you evaluate the function to determine the height of each rectangle, you get different types of sums:

• Left Riemann Sum: Uses the left endpoint of each subinterval for the height.
• Right Riemann Sum: Uses the right endpoint of each subinterval for the height.
• Midpoint Riemann Sum: Uses the exact middle $x$-value of each subinterval for the height.

Example: Right Riemann Sum

Problem: Approximate $\int_0^2 x^2 \, dx$ using $4$ rectangles with right endpoints.

1. Find $\Delta x$: $\Delta x = \frac{2 - 0}{4} = 0.5$
2. Identify the subintervals: $[0, 0.5], [0.5, 1], [1, 1.5], [1.5, 2]$
3. Find the heights: Evaluate $f(x) = x^2$ at the right endpoints ($0.5, 1, 1.5, 2$).
4. Calculate the total area: $\text{Area} \approx \Delta x [f(0.5) + f(1) + f(1.5) + f(2)]$ $\text{Area} \approx 0.5 [0.5^2 + 1^2 + 1.5^2 + 2^2]$ $\text{Area} \approx 0.5 [0.25 + 1 + 2.25 + 4] = 0.5 [7.5] = 3.75$

The Trapezoidal Rule

Instead of flat-topped rectangles, you can use trapezoids to better follow the natural slant of the curve.

The general formula for the Trapezoidal Rule with $n$ subintervals is: $T_n = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + \dots + 2f(x_{n-1}) + f(x_n)]$ Notice that the first and last endpoints are evaluated once, while all the inner endpoints are multiplied by $2$.

Example: Trapezoidal Rule

Problem: Estimate $\int_1^3 \frac{1}{x} \, dx$ using $n = 4$.

1. Find $\Delta x$: $\Delta x = \frac{3 - 1}{4} = 0.5$
2. Identify the endpoints: $x_0=1, x_1=1.5, x_2=2, x_3=2.5, x_4=3$
3. Apply the formula: $T_4 = \frac{0.5}{2} \left[ f(1) + 2f(1.5) + 2f(2) + 2f(2.5) + f(3) \right]$ $T_4 = 0.25 \left[ 1 + 2\left(\frac{2}{3}\right) + 2\left(\frac{1}{2}\right) + 2\left(\frac{2}{5}\right) + \frac{1}{3} \right]$ $T_4 = 0.25 \left[ 1 + \frac{4}{3} + 1 + \frac{4}{5} + \frac{1}{3} \right]$ $T_4 = 0.25 \left[ 2 + \frac{5}{3} + \frac{4}{5} \right] = 0.25 \left[ \frac{30 + 25 + 12}{15} \right] = 0.25 \left[ \frac{67}{15} \right] = \frac{67}{60} \approx 1.1167$

Connection to Definite Integrals

Approximations are just the beginning. As you increase the number of rectangles ($n \to \infty$), the width of each rectangle ($\Delta x$) shrinks to zero. The approximation gets closer and closer to the exact area.

This limit is the very definition of the definite integral: $\lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x = \int_a^b f(x) \, dx$