Problem 13 - 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. $$ f(x)=2 x \text { from } a=1 \text { to } b=2 $$

Step-by-Step Solution

Answer

Compute Riemann sums for \( n=10,100,1000 \) using a calculator, then integrate \( \int_{1}^{2} 2x \, dx \) to find the exact area.

Step 1: Define the problem

We are tasked with calculating Riemann sums for the function \( f(x) = 2x \) within the interval \([1, 2]\) using different partition counts \( n = 10, 100, \) and \( 1000 \). We need to use left, right, or midpoint rectangles and round the answers to three decimal places. We also aim to find the exact area under the curve using definite integration.

Step 2: Use a graphing calculator program for Riemann Sums

Use a graphing calculator or an online Riemann Sum calculator to find the approximate areas for \( n = 10, 100, \) and \( 1000 \) using left, right, or midpoint rectangles over the interval \([1, 2]\) for \( f(x) = 2x \). Input the function, interval, and values of \( n \) one by one to obtain the Riemann sums.

Step 3: Calculate Riemann sums for different n values

Based on the function and the chosen method (left, right, or midpoint), record the Riemann sums for different \( n \) values: \( n = 10, n = 100, \) and \( n = 1000 \). Ensure the results are rounded to three decimal places for consistency.

Step 4: Evaluate the definite integral

To find the exact area under the curve of \( f(x) = 2x \) from \( a = 1 \) to \( b = 2 \), compute the definite integral \( \int_{1}^{2} 2x \, dx \). This integral represents the exact area under the curve.

Step 5: Compare results

The Riemann sum results for different \( n \) values should approach the exact value you found through integration as \( n \) increases, confirming the accuracy of the Riemann sum approximation.

Key Concepts

Definite Integral

The definite integral of a function describes the accumulation of quantities over an interval. It quantifies the exact area under a curve between two points on the x-axis.

To calculate the definite integral of a function, we find:

• An antiderivative (a function whose derivative equals the original function). This involves integration.
• The difference between the values of the antiderivative evaluated at the upper and lower bounds of the interval.

For instance, if we want to find the definite integral of the function \( f(x) = 2x \) over the interval \([1, 2]\), we calculate \( \int_{1}^{2} 2x \, dx \). This integral gives us the exact area under the curve of \( f(x) = 2x \) from \( x = 1 \) to \( x = 2 \). Such calculations help us find precise solutions that approximate techniques like Riemann sums approach as they become refined with more subintervals.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a pivotal link between differentiation and integration. It states two main things:

• First, if a function is continuous over an interval, an antiderivative exists for it over the same interval.
• Second, the definite integral of a function over an interval can be computed using any one of its antiderivatives.

To apply this theorem when given a function such as \( f(x) = 2x \), you would:
- Find an antiderivative, for instance, \( F(x) = x^2 \).
- Use the limits of integration: evaluate \( F(x) \) at the upper bound \( b = 2 \) and the lower bound \( a = 1 \), then subtract: \( F(2) - F(1) = (2^2) - (1^2) = 4 - 1 = 3 \).

Thus, the Fundamental Theorem of Calculus allows us to calculate the definite integral, simplifying the process of finding the exact area under a curve.

Graphing Calculator

A graphing calculator is a powerful tool used to visualize functions, conduct numerical calculations, and solve complex mathematical problems, including Riemann sums and integrals.

When conducting Riemann Sum calculations, a graphing calculator can:

• Easily input the function and define the interval.
• Select the method (left, right, midpoint) for creating rectangles under the curve.
• Automate the computation for various \( n \) values, such as 10, 100, or 1000, facilitating faster and more accurate approximations.

By using this technology, students can learn effectively by comparing approximations with the exact results found using definite integrals. This reinforces the understanding of how increasing the number of rectangles in a Riemann sum leads to an approximation closer to the exact value given by the integral. Graphing calculators serve as both educational and practical resources to handle complex mathematical concepts with ease and precision, making them essential tools in learning and applying calculus.