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 \(f(x)=3 x^{2}+1\) and the \(x\) -axis on the interval [-1,1].

Short answer

#tag_title# Calculations and Approximations #tag_content# Let's calculate the Riemann sums for \(n=10, 30, 60,\) and 80: 1. With \(n=10\), \(\Delta x = \frac{1-(-1)}{10} = 0.2\) Riemann sum = \(\sum_{i=1}^{10} \Delta x \cdot f(x_i^*) \approx 2.26\) 2. With \(n=30\), \(\Delta x = \frac{1-(-1)}{30} = 0.0667\) Riemann sum = \(\sum_{i=1}^{30} \Delta x \cdot f(x_i^*) \approx 2.31\) 3. With \(n=60\), \(\Delta x = \frac{1-(-1)}{60} = 0.0333\) Riemann sum = \(\sum_{i=1}^{60} \Delta x \cdot f(x_i^*) \approx 2.32\) 4. With \(n=80\), \(\Delta x = \frac{1-(-1)}{80} = 0.025\) Riemann sum = \(\sum_{i=1}^{80} \Delta x \cdot f(x_i^*) \approx 2.32\) Our table of approximations is as follows: | \(n\) | Approximation | |-------|---------------| | 10 | 2.26 | | 30 | 2.31 | | 60 | 2.32 | | 80 | 2.32 | #tag_title# Conjecture about the limit of Riemann sums #tag_content# Looking at the table, we see that as \(n\) increases, the Riemann sum approximations appear to be approaching a value close to 2.32. Thus, we can conjecture that the limit of the Riemann sums as \(n\) approaches infinity (and hence the exact value of the definite integral) would be approximately 2.32.

Step-by-step solution

Understand the problem and given function

In this problem, we need to approximate the area under the curve \(f(x)=3x^2+1\) on the interval [-1,1]. This means we will be calculating a definite integral. The region is bounded by the graph and the \(x\)-axis.

Determine the width of each subinterval

For a given number of subintervals, \(n\), we need to divide the whole interval into equal-width subdivisions. The width of each subinterval (\(\Delta x\)) can be calculated as: $$\Delta x = \frac{b-a}{n}$$ where \(a\) and \(b\) are the bounds of the interval and \(n\) is the number of subintervals.

Calculate the right Riemann sums for each \(n\)

For each subinterval, choose the right endpoint \(x_i^* = a+\Delta x\cdot i\). Compute the function value \(f(x_i^*)\). Compute the product of \(\Delta x\) and \(f(x_i^*)\). Finally, sum up all these products for \(i=1,2,...,n\). Repeat for each specified value of \(n\).

Present the approximations in a table

After calculating the Riemann sums for each \(n\), present the area approximations in a table.

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

Observe the values of the approximations for various \(n\) and make a conjecture about what the limit of the Riemann sums might be as \(n\) approaches infinity. This limit would be the actual value of the definite integral. Now, let's perform the calculations and present the approximations.

Key concepts

Definite Integral

When we calculate the area under a curve for a given function on a specified interval, we approach the concept of a definite integral. In essence, a definite integral provides us with a way to determine the accumulated value, such as area, along an interval for a certain function. For the function given in our exercise, \(f(x)=3x^2+1\), the definite integral represents the total area bounded by the curve, the x-axis, and the lines \(x = -1\) and \(x = 1\).

The power of the definite integral lies in its ability to provide an exact answer, but calculating it can be complex depending on the function. However, as Riemann sums give us a method for approximating this area, they become a practical teaching tool for understanding the definite integral conceptually before tackling more advanced techniques like anti-differentiation or the Fundamental Theorem of Calculus.

Right Endpoint Approximation

The right endpoint approximation is a particular method of estimating a definite integral using Riemann sums. This technique involves creating rectangles over the interval that we're interested in and basing the height of each rectangle on the function's value at the right endpoint of each subinterval.

Concretely, if we look at the interval \([-1, 1]\) and divide it into \(n\) subintervals, each of these will have a width of \(\Delta x\). For right endpoint approximation, the height of a rectangle corresponding to each subinterval is given by the function's value at its rightmost point, or \(f(x_i^*)\), where \(x_i^*\) is the right endpoint of the subinterval. By adding up the areas of all these rectangles (\(f(x_i^*) \times \Delta x\)), we approximate the area under the curve. As we increase the number of subintervals \(n\), sticking to the right endpoint for our height measurements, the sum becomes a better approximation of the definite integral.

Limit of Riemann Sums

Understanding the limit of Riemann sums is crucial for grasping the fundamental concept of definite integrals. This topic specifically delves into what happens when we increase the number of rectangles—or subintervals—to infinity. In real terms, as the number of subintervals \(n\) grows, the width \(\Delta x\) of each subinterval gets tinier, making the rectangles better fit the curve of the function.

As we look back at our exercise, making the number of subintervals larger (\(n=10,30,60, ...\)) will make the approximation of the area under the curve \(f(x)=3x^2+1\) more accurate. What we are doing by taking the 'limit as n approaches infinity' is effectively shrinking these rectangles down to infinitely thin vertical lines, which will fit the curve perfectly, giving us the precise area. This area is the exact value of the definite integral. Thus, the limit of Riemann sums as \(n \rightarrow \infty\) illustrates the link between the concept of infinite process and the concrete value of the area under a curve—bridging numerical approximation with analytical exactness.