Left Riemann Sums: Worked Examples and Practice Problems

Understanding Left Riemann Sums 🧐

The left Riemann sum is a method for approximating the definite integral of a function. It uses rectangles to estimate the area under the curve, where the height of each rectangle is determined by the function's value at the left endpoint of each subinterval.

Formula for Left Riemann Sums 📝

Given a function $f(x)$ on the interval $[a, b]$, we divide the interval into $n$ equal subintervals of width $\Delta x = \frac{b - a}{n}$. The left Riemann sum is then given by:

$\qquad L_n = \Delta x \left[ f(x_0) + f(x_1) + \cdots + f(x_{n-1}) \right]$

where $x_i = a + i \Delta x$ for $i = 0, 1, \dots, n-1$.

Worked Example 1: Approximating $\int_1^4 x^2 dx$ using $n=3$ subintervals 🚀

Let's approximate the definite integral $\int_1^4 x^2 dx$ using a left Riemann sum with $n=3$ subintervals.

1. Calculate $\Delta x$: $\Delta x = \frac{4 - 1}{3} = 1$.
2. Determine the left endpoints: $x_0 = 1$, $x_1 = 2$, $x_2 = 3$.
3. Evaluate the function at the left endpoints: $f(1) = 1^2 = 1$, $f(2) = 2^2 = 4$, $f(3) = 3^2 = 9$.
4. Apply the formula: $L_3 = 1 \cdot (1 + 4 + 9) = 14$.

Thus, the left Riemann sum approximation is 14.

Code Implementation (Python) 🐍

def left_riemann_sum(f, a, b, n):
    delta_x = (b - a) / n
    x_values = [a + i * delta_x for i in range(n)]
    return delta_x * sum(f(x) for x in x_values)

def f(x):
    return x**2

a = 1
b = 4
n = 3

approximation = left_riemann_sum(f, a, b, n)
print(f"Left Riemann Sum Approximation: {approximation}")

Worked Example 2: Approximating $\int_0^2 e^x dx$ using $n=4$ subintervals 💡

Approximate the definite integral $\int_0^2 e^x dx$ using a left Riemann sum with $n=4$ subintervals.

1. Calculate $\Delta x$: $\Delta x = \frac{2 - 0}{4} = 0.5$.
2. Determine the left endpoints: $x_0 = 0$, $x_1 = 0.5$, $x_2 = 1$, $x_3 = 1.5$.
3. Evaluate the function at the left endpoints: $f(0) = e^0 = 1$, $f(0.5) = e^{0.5} \approx 1.6487$, $f(1) = e^1 \approx 2.7183$, $f(1.5) = e^{1.5} \approx 4.4817$.
4. Apply the formula: $L_4 = 0.5 \cdot (1 + 1.6487 + 2.7183 + 4.4817) = 0.5 \cdot 10.8487 \approx 5.4244$.

The left Riemann sum approximation is approximately 5.4244.

Practice Problems 💪

• Approximate $\int_0^1 x^3 dx$ using a left Riemann sum with $n=5$ subintervals.
• Approximate $\int_1^3 \frac{1}{x} dx$ using a left Riemann sum with $n=4$ subintervals.

Common Mistakes to Avoid ⚠️

• Forgetting to multiply by $\Delta x$.
• Using right endpoints instead of left endpoints.
• Incorrectly calculating the function values.

Further Resources 📚

• Khan Academy: Riemann Sums
• Calculus textbooks for additional examples and explanations.