Problem 16 - Applied Calculus

Question

Use a graphing calculator Riemann Sum program from the Internet (see page 332 ) to find the following Riemann sums. i. Calculate the Riemann sum for each function for the following values of \(n: 10,100,\) and \(1000 .\) Use left, right, or midpoint rectangles, making a table of the answers, rounded to three decimal places. ii. Find the exact value of the area under the curve by evaluating an appropriate definite integral using the Fundamental Theorem. The values of the Riemann sums from part (i) should approach this number. $$ f(x)=e^{x} \text { from } a=-1 \text { to } b=1 $$

Step-by-Step Solution

Answer

Use midpoint Riemann sums for \(n = 10, 100, 1000\). Calculate exact area as \(e - \frac{1}{e}\). Riemann sums approach integral value with increasing \(n\).

Step 1: Understand the Problem

We need to calculate Riemann sums for the function \(f(x) = e^x\) over the interval \([-1, 1]\) using different numbers of rectangles: \(n = 10, 100,\) and \(1000\). We will use left, right, or midpoint Riemann sums and round the results to three decimal places. In part (ii), we will find the exact area using a definite integral.

Step 2: Choose Riemann Sum Method

We will use the midpoint Riemann sum for this exercise, as it often provides a balance between the overestimation and underestimation. We will use the midpoint of each subinterval to evaluate the function \(f(x) = e^x\).

Step 3: Calculate Riemann Sums for n=10

Divide the interval \([-1, 1]\) into \(n=10\) subintervals, each of width \(\Delta x = \frac{1-(-1)}{10} = 0.2\). Compute the midpoint \(x_i^*\) for each subinterval and then calculate the Riemann sum: \[ R_{10} = \sum_{i=1}^{10} e^{x_i^*} \Delta x \] Compute this using a calculator or program and round to three decimal places.

Step 4: Calculate Riemann Sums for n=100

Divide the interval \([-1, 1]\) into \(n=100\) subintervals, each of width \(\Delta x = 0.02\). Calculate the midpoint \(x_i^*\) for each subinterval and compute the Riemann sum: \[ R_{100} = \sum_{i=1}^{100} e^{x_i^*} \Delta x \] Use a calculator or program to compute this, rounding to three decimal places.

Step 5: Calculate Riemann Sums for n=1000

Divide the interval \([-1, 1]\) into \(n=1000\) subintervals, each of width \(\Delta x = 0.002\). Calculate the midpoint \(x_i^*\) for each subinterval and compute the Riemann sum: \[ R_{1000} = \sum_{i=1}^{1000} e^{x_i^*} \Delta x \] Use a calculator or program to compute this, rounding to three decimal places.

Step 6: Calculate the Definite Integral

Using the Fundamental Theorem of Calculus, evaluate the exact area under the curve from \(-1\) to \(1\) by calculating the definite integral: \[ \int_{-1}^{1} e^x \, dx \] Find the antiderivative of \(e^x\), which is \(e^x\). Evaluate \[ \left[ e^x \right]_{-1}^{1} = e^1 - e^{-1} = e - \frac{1}{e} \] Calculate this difference and round it to three decimal places.

Step 7: Compare Results

Compare the Riemann sums for \(n=10, 100,\) and \(1000\) with the exact integral value obtained. As \(n\) increases, the Riemann sums should approach the exact integral value.

Key Concepts

Midpoint Rule

The Midpoint Rule is a method for approximating the total area under a curve. It is a specific type of Riemann sum that uses the midpoint of each subinterval to determine the height of the rectangles that approximate the area. By dividing the interval into smaller subintervals, and evaluating the function at the midpoint \(x_i^*\) of these subintervals, we can get a more accurate estimate of the integral.
To apply the Midpoint Rule:

• Divide the interval \([-1, 1]\) into equally spaced subintervals, with width \(\Delta x\).
• Calculate the midpoint of each subinterval.
• Evaluate the function at these midpoints, multiply by \(\Delta x\), and sum the results.

This method helps reduce overestimation or underestimation. As the number of subintervals \(n\) increases, the approximation becomes more precise. For example, reaching \(n = 1000\), the result converges closely to the exact integral value.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the gap between differentiation and integration. It states that if a function is continuous on a closed interval \([a, b]\), then the integral of the function over that interval is the difference of its antiderivative evaluated at the endpoints.
For the function \(f(x) = e^x\), its antiderivative is also \(e^x\). To find the exact value of the area under the curve from \(-1\) to \(1\), you calculate:

• Find the antiderivative: \(\left[ e^x \right]\).
• Evaluate it at the upper and lower limits: \(e^1 - e^{-1}\).
• This produces the exact area under the curve.

The Riemann sums approximate this value and approach it as \(n\) increases.

Definite Integral

A definite integral calculates the net area under a curve from one point to another. This is a crucial concept in calculus, providing the exact measurement of the area, unlike Riemann sums, which approximate it.
Performing a definite integral involves:

• Identifying the function to be integrated \(f(x) = e^x\).
• Determining the interval \([a, b] = [-1, 1]\).
• Finding the antiderivative \(F(x) = e^x\).
• Evaluating the antiderivative at the boundary points: \(F(1) - F(-1) = e - \frac{1}{e}\).

This result is the exact area the Midpoint Rule and other Riemann sums are approximating. Riemann sums become more accurate as the number of subintervals increases, converging toward the definite integral's value.

Antiderivative

An antiderivative of a function is another function whose derivative is the original function. In this exercise, we are interested in the antiderivative for \(f(x) = e^x\), which is straightforwardly \(e^x\) itself.
Finding antiderivatives is essential for evaluating definite integrals, as it allows us to apply the Fundamental Theorem of Calculus. Here’s why antiderivatives are important:

• They help in determining areas under curves exactly.
• Once found, allow direct computation of the definite integrals.
• They are used to compute the difference at the limits \(F(b) - F(a)\).

For this problem, calculating the definite integral involves finding the difference between the antiderivative evaluated at the interval bounds, yielding the area captured exactly as \(e - \frac{1}{e}\). As \(n\) increases in the Riemann sums, the summations approximate this exact result more closely.