Problem 17 - 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. $$ f(x)=\frac{1}{x} \text { from } a=1 \text { to } b=2 $$

Step-by-Step Solution

Answer

The Riemann sums approach the definite integral value of \( \ln(2) \approx 0.693 \).

Step 1: Define the Function and Interval

We are given the function \[ f(x) = \frac{1}{x} \]which is to be evaluated from \( a = 1 \) to \( b = 2 \).

Step 2: Calculate Riemann Sums Using Left Rectangles

For left rectangles, use the formula:\[ \text{Riemann sum} = \sum_{i=0}^{n-1} f(x_i^*) \Delta x \]where \( x_i^* = a + i\Delta x \), and \( \Delta x = \frac{b-a}{n} \). Calculate this for \( n = 10, 100, 1000 \).

Step 3: Calculate Riemann Sums Using Right Rectangles

For right rectangles, the point \( x_i^* = a + (i+1)\Delta x \).Use the same formula and calculate this for \( n = 10, 100, 1000 \).

Step 4: Calculate Riemann Sums Using Midpoint Rectangles

For midpoint rectangles, the point \( x_i^* = a + \left(i + \frac{1}{2}\right)\Delta x \). Use the formula and calculate this for \( n = 10, 100, 1000 \).

Step 5: Compile Riemann Sum Results into a Table

Create a table summarizing the Riemann sums for each method (Left, Right, Midpoint) and each \( n \) (10, 100, 1000), rounded to three decimal places.

Step 6: Set Up and Evaluate the Definite Integral

The exact area under the curve can be found by evaluating the definite integral:\[ \int_{1}^{2} \frac{1}{x} \, dx \]Using the Fundamental Theorem of Calculus, find:\[ \ln|x| \bigg|_{1}^{2} = \ln(2) - \ln(1) = \ln(2) \]

Step 7: Compare Riemann Sums to the Exact Value

Compare the Riemann sums from the table to the exact value calculated by the integral. As \( n \) increases, the Riemann sums should converge to the exact value of approximately 0.693.

Key Concepts

Definite Integral

Understanding a definite integral is crucial to finding the area under a curve. It is essentially the limit of a sum of areas of rectangles beneath or above a curve. For the given function \[ f(x) = \frac{1}{x} \] within the interval from \( a = 1 \) to \( b = 2 \), the goal is to use a definite integral to find the exact area between the curve and the x-axis.

• The definite integral is represented by \( \int_{a}^{b} f(x) \, dx \).
• It calculates the net area, accounting for sections where the curve lies both above and below the x-axis.
• For our exercise, the computation leads to \( \int_{1}^{2} \frac{1}{x} \, dx \), which can be solved using the properties of logarithmic functions.

Evaluating this leads us to the exact value of \( \ln(2) \)—a crucial point of comparison for the Riemann sums.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus serves as a bridge between integration and differentiation, providing a method to evaluate definite integrals more easily. It states that if \( F(x) \) is an antiderivative of \( f(x) \), then the definite integral of \( f(x) \) from \( a \) to \( b \) is:\[ F(b) - F(a) \]

• In this context, the antiderivative of \( f(x) = \frac{1}{x} \)is \( F(x) = \ln|x| \).
• Applying the theorem, we compute \( \ln(2) - \ln(1) \), resulting in \( \ln(2) \).
• This value, approximately 0.693, shows us the exact area under the curve from \( x = 1 \) to \( x = 2 \).

The Riemann sums calculated through various methods should approach this exact value, illustrating the theorem in action.

Midpoint Rule

The midpoint rule is one method for approximating the area under a curve using Riemann sums. This method employs rectangles whose heights are determined by the function value at the midpoint of each subinterval.

• The point \( x_i^* \)in this case is given by \( a + \left(i + \frac{1}{2}\right)\Delta x \).
• This approach often provides better approximations than the left or right endpoint methods, especially with fewer subintervals (lower \( n \)).
• In the exercise, the midpoint rule is applied for \( n = 10, 100, 1000 \), progressively refining the sum to converge to the exact value of the definite integral.

When computing these midpoints, the results offer an insightful view of how increasing \( n \)improves the estimation of the integral, showcasing the efficiency of this rule.

Left and Right Rectangles

In approximating definite integrals, left and right rectangles provide two simple techniques for Riemann sums. These methods calculate the sum of areas using rectangles formed by evaluating the function at either the start or end of the subintervals.

• **Left Rectangles:** The height of each rectangle is determined using the function value at the left endpoint \( x_i^* = a + i\Delta x \).
• **Right Rectangles:** Here, the function is evaluated at the right endpoint \( x_i^* = a + (i+1)\Delta x \).
• Both methods are applied in the exercise for \( n = 10, 100, 1000 \)).As \( n \) increases, both the left and right Riemann sums converge towards the true value of the integral.

These two approaches help visualize how the choice of approximation influences the sum, highlighting the impact of subinterval counts and endpoint selection on accuracy.