Problem 46 - Calculus Early Transcendentals

Question

Riemann sums for larger values of \(n\) Complete the following steps for the given function \(f\) and interval. a. For the given value of \(n\), use sigma notation to write the left, right, and midpoint Riemann sums. Then evaluate each sum using a calculator. b. Based on the approximations found in part (a), estimate the area of the region bounded by the graph of \(f\) and the \(x\) -axis on the interval. $$f(x)=\cos 2 x \text { on }[0, \pi / 4] ; n=60$$.

Step-by-Step Solution

Answer

Question: Estimate the area of the region bounded by the graph of \(f(x) = \cos(2x)\) and the x-axis on the interval \([0, \frac{\pi}{4}]\) using \(n = 60\). Answer: The area can be estimated using the formula \(Area \approx \frac{L_n + R_n + M_n}{3}\). Calculate the values of the left, right, and midpoint Riemann sums \(L_n\), \(R_n\), and \(M_n\) using a calculator. Then plug the values into the formula and evaluate to get the approximate area of the region.

Step 1: Determine the width of each subinterval (delta x)

\(\Delta x = \frac{b - a}{n} = \frac{\frac{\pi}{4} - 0}{60} = \frac{\pi}{240}\).

Step 2: Write the left Riemann sum using sigma notation

The left Riemann sum is given by: $$L_n = \sum_{i=0}^{n-1} f(a + i\Delta x)\Delta x = \sum_{i=0}^{59} \cos(2(0 + i\frac{\pi}{240}))\frac{\pi}{240}$$.

Step 3: Write the right Riemann sum using sigma notation

The right Riemann sum is given by: $$R_n = \sum_{i=1}^{n} f(a + i\Delta x)\Delta x = \sum_{i=1}^{60} \cos(2(0 + i\frac{\pi}{240}))\frac{\pi}{240}$$.

Step 4: Write the midpoint Riemann sum using sigma notation

The midpoint Riemann sum is given by: $$M_n = \sum_{i=0}^{n-1} f(a + (i + \frac{1}{2})\Delta x)\Delta x = \sum_{i=0}^{59} \cos(2(0 + (i+\frac{1}{2})\frac{\pi}{240}))\frac{\pi}{240}$$.

Step 5: Evaluate the Riemann sums using a calculator

At this stage, you can use a calculator to compute the values of the Riemann sums \(L_n\), \(R_n\), and \(M_n\). After computing these values, proceed to the final step to estimate the area of the region.

Step 6: Estimate the area of the region

The area of the region bounded by the graph of \(f(x)=\cos(2x)\) and the x-axis on the interval \([0, \frac{\pi}{4}]\) can be approximated by the average of the left, right, and midpoint Riemann sums: $$Area \approx \frac{L_n + R_n + M_n}{3}$$. Using the calculator, find the area by plugging in the values of \(L_n\), \(R_n\), and \(M_n\).

Key Concepts

Sigma Notation

Sigma notation, represented by the Greek capital letter sigma () is a concise and powerful way of expressing the summation of a sequence of terms. This notation becomes particularly useful when dealing with a large number of terms, which is common in calculus where we add up slices to approximate areas, volumes, and other quantities.

In a sigma notation expression, the variable below the sigma symbol indicates where the summation starts, known as the lower limit, while the number or expression above denotes where it ends, called the upper limit. The term to the right of the sigma describes the value to be added at each step, often as a function of the variable.

To give a better understanding using the Riemann sum example, the notation would be read as 'the sum of from i equals 0 to 59.' Here, the represents the height of each rectangle, and is the width of each subinterval in the context of the left Riemann sum. Sigma notation provides a method for calculating the sum of values generated by a rule, quickly and easily, especially when used with a calculator or computer.

Integral Approximation

Integral approximation is a central concept in calculus, used to estimate the value of a definite integral. It is rooted in the problem of finding the area under a curve and can be visualized as summing the areas of shapes (like rectangles or trapezoids) that approximate the region below the curve.

The Riemann sums, including left, right, and midpoint, are methods of integral approximation. Each of them uses rectangles to approximate the area under the curve, and the sum of the areas of these rectangles approaches the true value of the integral as the number of rectangles increases, that is, as n becomes larger.

Understanding Delta x ()

In the example given, , which represents the width of each rectangle, becomes smaller as increases, leading to a more accurate approximation.

Left, Right, and Midpoint Approaches

The left Riemann sum takes the height of the rectangle from the left end of the subinterval, the right Riemann sum from the right end, and the midpoint Riemann sum from the value at the midpoint of the subinterval.

Using calculus, the true area under the curve is found by taking the limit of these sums as tends to infinity, which coincides with the concept of the definite integral.

Area Under a Curve

The concept of the 'area under a curve' is a foundational block of integral calculus. It refers to the area bounded by the graph of a function, the x-axis, and the vertical lines corresponding to the limits of integration. Mathematically, it can be represented as a definite integral.

The exercise involves approximating the area under the curve of the function using Riemann sums. The true area under a curve is a limit of these approximations as the subdivisions turn infinitely small—the more subdivisions used, the closer the approximation to the actual area.

The simplest geometric shapes used for approximation are rectangles (Riemann sums), but other shapes like trapezoids (Trapezoidal Rule) and parabolas (Simpson's Rule) can also be used for better accuracy. By computing the left (), right (), and midpoint ( sums and averaging them, we obtain a better approximation of the true area under the function over the interval from 0 to .

This method of approximating the area is incredibly useful in real-world applications where exact calculations are difficult or impossible to perform, such as calculating the distance traveled by a vehicle over time with variable speed or the total energy consumed by an appliance with variable power intake over a period.