Right Riemann Sums: Calculus Exam Practice Problems Solved

Right Riemann Sums Explained ➕

The right Riemann sum is a method for approximating the definite integral of a function. It involves dividing the interval of integration into subintervals and using the right endpoint of each subinterval to determine the height of the rectangle. The sum of the areas of these rectangles approximates the definite integral.

Formula

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

$$R_n = \sum_{i=1}^{n} f(x_i) \Delta x$$

where $x_i = a + i \Delta x$ is the right endpoint of the $i$-th subinterval.

Example Problem 1 🚀

Approximate the definite integral $\int_{1}^{4} x^2 dx$ using a right Riemann sum with $n = 3$ subintervals.

Solution

1. Calculate $\Delta x$:

$$\Delta x = \frac{4-1}{3} = \frac{3}{3} = 1$$

2. Determine the right endpoints:

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

3. Evaluate the function at the right endpoints:

$f(2) = 2^2 = 4$, $f(3) = 3^2 = 9$, $f(4) = 4^2 = 16$

4. Calculate the right Riemann sum:

$$R_3 = \sum_{i=1}^{3} f(x_i) \Delta x = f(2) \cdot 1 + f(3) \cdot 1 + f(4) \cdot 1 = 4 + 9 + 16 = 29$$

Thus, the right Riemann sum approximation is 29.

Example Problem 2 💡

Approximate the definite integral $\int_{0}^{2} (x^3 + 1) dx$ using a right Riemann sum with $n = 4$ subintervals.

Solution

1. Calculate $\Delta x$:

$$\Delta x = \frac{2-0}{4} = \frac{2}{4} = 0.5$$

2. Determine the right endpoints:

$x_1 = 0 + 0.5 = 0.5$, $x_2 = 0.5 + 0.5 = 1$, $x_3 = 1 + 0.5 = 1.5$, $x_4 = 1.5 + 0.5 = 2$

3. Evaluate the function at the right endpoints:

$f(0.5) = (0.5)^3 + 1 = 1.125$, $f(1) = 1^3 + 1 = 2$, $f(1.5) = (1.5)^3 + 1 = 4.375$, $f(2) = 2^3 + 1 = 9$

4. Calculate the right Riemann sum:

$$R_4 = \sum_{i=1}^{4} f(x_i) \Delta x = (1.125 + 2 + 4.375 + 9) \cdot 0.5 = 16.5 \cdot 0.5 = 8.25$$

Thus, the right Riemann sum approximation is 8.25.

Code Example 💻

Here's a Python code snippet to calculate the right Riemann sum:

def right_riemann_sum(f, a, b, n):
    delta_x = (b - a) / n
    right_sum = 0
    for i in range(1, n + 1):
        x_i = a + i * delta_x
        right_sum += f(x_i) * delta_x
    return right_sum

# Example usage:
def f(x):
    return x**2

a = 1
b = 4
n = 3

result = right_riemann_sum(f, a, b, n)
print(result)  # Output: 29.0

Key Takeaways ✨

• Right Riemann sums approximate definite integrals using rectangles.
• The accuracy of the approximation improves as the number of subintervals ($n$) increases.
• Understanding and applying the formula correctly is crucial for solving problems.