Math Problem Statement

Evaluate the left and right Riemann sums for f over the interval [0, 2] with n = 4 using the provided table of values for f(x).

Solution

The problem asks to evaluate the left and right Riemann sums for ff over the interval [0,2][0, 2] with n=4n = 4 using the provided table of values.

The given dataThe problem asks to evaluate the left and right Riemann sums for ff over the interval [0,2][0, 2] with n=4n = 4 using the provided table of values.

The given

  • Interval: [0,2][0, 2]
  • Number of subintervals: n=4n = 4
  • Values for xx and f(x)f(x):
\hline x & 0 & 0.5 & 1 & 1.5 & 2 \\ \hline f(x) & 5 & 3 & 2 & 1 & 1 \\ \hline \end{array}$$ ### Step 1: Determine the width of each subinterval The width $$\Delta x$$ of each subinterval is calculated as: $$\Delta x = \frac{b - a}{n} = \frac{2 - 0}{4} = 0.5$$ ### Step 2: Left Riemann sum For the **left Riemann sum**, we use the left endpoint of each subinterval. The left endpoints are $$x = 0, 0.5, 1, 1.5$$, and the corresponding values of $$f(x)$$ are 5, 3, 2, and 1. The left Riemann sum is: $$L_4 = \Delta x \left( f(0) + f(0.5) + f(1) + f(1.5) \right)$$ $$L_4 = 0.5 \times (5 + 3 + 2 + 1) = 0.5 \times 11 = 5.5$$ ### Step 3: Right Riemann sum For the **right Riemann sum**, we use the right endpoint of each subinterval. The right endpoints are $$x = 0.5, 1, 1.5, 2$$, and the corresponding values of $$f(x)$$ are 3, 2, 1, and 1. The right Riemann sum is: $$R_4 = \Delta x \left( f(0.5) + f(1) + f(1.5) + f(2) \right)$$ $$R_4 = 0.5 \times (3 + 2 + 1 + 1) = 0.5 \times 7 = 3.5$$ ### Final Results: - Left Riemann sum: $$5.5$$ - Right Riemann sum: $$3.5$$ Would you like further details or clarifications on the process? --- Here are five related questions to explore the concept further: 1. How does increasing the number of subintervals $$n$$ affect the accuracy of the Riemann sums? 2. What is the significance of using the left versus right Riemann sums in estimating area under a curve? 3. How would you compute the midpoint Riemann sum for the same data? 4. Can you derive the actual integral of $$f(x)$$ over $$[0, 2]$$ using a known function for $$f(x)$$? 5. How do Riemann sums relate to the concept of definite integrals? **Tip**: The more subintervals you use (larger $$n$$), the more accurate your Riemann sum approximation becomes for the actual integral!

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Math Problem Analysis

Mathematical Concepts

Riemann Sums
Definite Integrals
Numerical Approximation

Formulas

Left Riemann sum formula: L_n = Δx * (f(x_0) + f(x_1) + ... + f(x_{n-1}))
Right Riemann sum formula: R_n = Δx * (f(x_1) + f(x_2) + ... + f(x_n))
Δx = (b - a) / n

Theorems

Riemann sum theorem: As the number of subintervals n approaches infinity, the Riemann sum approaches the definite integral.

Suitable Grade Level

Grades 11-12 (Calculus)

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