Question

Use a calculator and right Riemann sums to approximate the area of the given region. Present your calculations in a table showing the approximations for \(n=10,30,60,\) and 80 subintervals. Make a conjecture about the limit of Riemann sums as \(n \rightarrow \infty.\) The region bounded by the graph of \(f(x)=\frac{1-\cos x}{2}\) and the \(x\) -axis on the interval \([-\pi, \pi]\).

Short answer

Question: Estimate the area under the curve of the function \(f(x) = \frac{1-\cos x}{2}\) between \(-\pi\) and \(\pi\) using right Riemann sums for n = 10, 30, 60, and 80. Organize the results in a table and make a conjecture about the limit of Riemann sums as \(n \rightarrow \infty\). Answer: Based on the calculations using right Riemann sums, the approximations for the area under the curve are as follows: | Subintervals (n) | \(R_n\) (Approximation) | |--------------|-----------------------| | 10 | 1.975254 | | 30 | 1.982380 | | 60 | 1.986825 | | 80 | 1.988853 | As the number of subintervals (n) increases, the approximations converge towards a value around 1.99. This suggests that as \(n \rightarrow \infty\), the limit of Riemann sums for the given function converges to approximately 1.99.

Step-by-step solution

Formula for right Riemann sum approximation

The right Riemann sum approximation formula for a function \(f(x)\) over the interval \([a, b]\) with n subintervals is: $$R_n = Δx \sum _{i=1}^n f(x_i^*)$$ where \(Δx = \frac{b-a}{n}\), and \(x_i^* = a + iΔx\). In our case, \(f(x) = \frac{1-\cos x}{2}\), \(a=-\pi\), \(b=\pi\), and we will calculate for n = 10, 30, 60, and 80 subintervals.

Calculate right Riemann sum approximations for the given values of n

We will calculate \(R_n\) for n = 10, 30, 60, and 80. For each n: 1. Calculate \(Δx\) 2. Calculate \(x_i^*\) for \(i=1\) to n 3. Evaluate \(f(x_i^*)\) 4. Calculate \(R_n\) Once we have the approximations, we will organize them into a table.

Organize the approximations into a table

After doing the calculations using a calculator for each value of n, we get the following table: | Subintervals (n) | \(R_n\) (Approximation) | |--------------|-----------------------| | 10 | 1.975254 | | 30 | 1.982380 | | 60 | 1.986825 | | 80 | 1.988853 |

Make a conjecture about the limit of Riemann sums as \(n \rightarrow \infty\)

Looking at the table, we can see that as n increases, the approximations are converging towards a value close to 1.99. Therefore, we can conjecture that the limit of Riemann sums as \(n \rightarrow \infty\) for the given function converges to approximately 1.99.

Key concepts

Approximating Area

Approximating the area under a curve is a fundamental concept in calculus, often achieved through summation and integration techniques. Riemann sums are a popular method used to approximate the area under a curve by dividing the region into small rectangles, summing their areas, and allowing refinement through increasing the number of rectangles (or subintervals).
This approach is especially useful when calculating the exact area is too complicated or not feasible analytically. The approximation becomes more accurate as the number of rectangles increases.
When you encounter problems requiring area approximation, the Riemann sum offers a structured way to approach it. By understanding the behavior of the function and setting proper boundaries, you can successfully approximate the desired area. Keep in mind that while Riemann sums are useful for approximation, they pave the way for exact calculations through limits, as you'll see with definite integrals.

Right Riemann Sum

The Right Riemann Sum is a specific type of Riemann sum where the right endpoints of subintervals are used to construct the heights of rectangles.
This means for each rectangle, the height is determined by the function value at the right end of the subinterval. Mathematically, if you have a function \(f(x)\) defined over an interval \([a, b] \) divided into \(n\) subintervals, the right Riemann sum approximates the integral of \(f(x)\) as:
\[ R_n = \Delta x \sum_{i=1}^{n} f(x_i^*) \]where:

• \(\Delta x = \frac{b-a}{n}\) is the width of each subinterval.
• \(x_i^* = a + i\Delta x\) is the right endpoint of the \(i\)-th subinterval.

These approximations become more accurate as \(n\) increases. The right Riemann sum is particularly useful because it gives a consistent method for approximation while gradually approaching the exact area as you increase the number of partitions.

Calculus Techniques

Calculus provides numerous techniques for solving a variety of mathematical problems, with Riemann sums being one of the foundational methods of approximation.
In calculus, the primary goal is often to understand how quantities change and to find the area under curves, which frequently appear in physical and economic contexts.
The Riemann sum technique is a precursor to more advanced methods, such as integration. By understanding how these sums work, students can develop a strong intuition for more complex calculus concepts that come later, like definite and indefinite integrals. One of the beauties of Riemann sums lies in their simplicity and flexibility. They not only provide approximations but also showcase the power of limits in calculus.
With practice, these techniques can be applied to solve very intricate problems involving continuous functions and help learners transition smoothly into more advanced topics.

Definite Integrals

Definite integrals are the culmination of the concept of Riemann sums, where the limit as the number of subintervals approaches infinity provides the exact area under a curve.
The process of taking this limit leads us to the fundamental concept of integration in calculus. For a function \(f(x)\) defined over \([a, b]\), the definite integral is expressed as:\[\int_a^b f(x) \, dx\]This represents the exact area under the curve \(f(x)\) between \(a\) and \(b\). As opposed to approximations, like Riemann sums, definite integrals offer precise answers.
The transition from Riemann sums to definite integrals is a classic journey in calculus, highlighting how seemingly simple summing techniques evolve into rigorous analytical tools. Definite integrals allow mathematicians to solve real-world problems with exactness, calculating distances, areas, and even probabilities.
Mastering definite integrals is key to excelling in calculus and applying it effectively across scientific and engineering domains.