Problem 31 - Calculus Single Variable

Question

In each of Exercises \(27-38\), calculate the right endpoint approximation of the area of the region that lies below the graph of the given function \(f\) and above the given interval \(I\) of the \(x\) -axis. Use the uniform partition of given order \(N\). $$ f(x)=2+\sin (2 x) \quad I=[\pi / 2,2 \pi], N=3 $$

Step-by-Step Solution

Answer

The approximate area under the curve is \(3\pi\).

Step 1: Determine the Interval Length

Given that the interval \(I = [\frac{\pi}{2}, 2\pi] \), the total length of the interval is \(2\pi - \frac{\pi}{2} = \frac{3\pi}{2}\). This is the total length over which we want to approximate the area.

Step 2: Calculate Subinterval Width

Divide the total interval length by the number of partitions \(N = 3\). The width of each subinterval, denoted \( \Delta x \), is: \[ \Delta x = \frac{\frac{3\pi}{2}}{3} = \frac{\pi}{2} \].

Step 3: Identify the Right Endpoints

For the given interval \([\frac{\pi}{2}, 2\pi]\), with subinterval width \( \frac{\pi}{2} \), the right endpoints are calculated as follows:- First interval right endpoint: \( \frac{\pi}{2} + \frac{\pi}{2} = \pi \)- Second interval right endpoint: \( \pi + \frac{\pi}{2} = \frac{3\pi}{2} \)- Third interval right endpoint: \( \frac{3\pi}{2} + \frac{\pi}{2} = 2\pi \).These endpoints will be used in evaluating the function for the approximation.

Step 4: Evaluate the Function at Right Endpoints

Compute \( f(x) = 2 + \sin(2x) \) for each right endpoint:- \( f(\pi) = 2 + \sin(2\cdot\pi) = 2 + \sin(2\pi) = 2 + 0 = 2 \)- \( f(\frac{3\pi}{2}) = 2 + \sin(2\cdot\frac{3\pi}{2}) = 2 + \sin(3\pi) = 2 + 0 = 2 \)- \( f(2\pi) = 2 + \sin(2\cdot2\pi) = 2 + \sin(4\pi) = 2 + 0 = 2 \).

Step 5: Calculate the Total Approximate Area

The approximate area using the right endpoint approximation and the sum of rectangular areas is calculated as:\[ A \approx \sum_{i=1}^{3} f(x_i) \cdot \Delta x = f(\pi) \cdot \frac{\pi}{2} + f(\frac{3\pi}{2}) \cdot \frac{\pi}{2} + f(2\pi) \cdot \frac{\pi}{2} \]\[ A \approx 2 \cdot \frac{\pi}{2} + 2 \cdot \frac{\pi}{2} + 2 \cdot \frac{\pi}{2} \]\[ A \approx \pi + \pi + \pi = 3\pi \]

Step 6: Conclusion: Final Answer

The right endpoint approximation of the area under the curve \(f(x)\) over the interval \(I\) is \(3\pi\).

Key Concepts

Subinterval Width

When approximating the area under a curve using right endpoint approximation, understanding the subinterval width is crucial.
The width of a subinterval, often denoted as \( \Delta x \), dictates how finely or coarsely we partition our interval. In this context:

• The interval \( I \) is given as \( \left[ \frac{\pi}{2}, 2\pi \right] \).
• The total length of this interval is \( 2\pi - \frac{\pi}{2} = \frac{3\pi}{2} \).
• With a partition of order \( N = 3 \), the subinterval width is calculated by dividing the interval length by the number of partitions.

This gives us:
\[ \Delta x = \frac{\frac{3\pi}{2}}{3} = \frac{\pi}{2} \]
Each resulting subinterval is \( \frac{\pi}{2} \) units wide. This consistent width allows for precise calculations and ensures accuracy across all subdivisions.

Riemann Sum

The Riemann sum is a method for approximating the total area under a curve on a graph, often represented by summing the areas of rectangles.
In this exercise, a right endpoint Riemann sum is used, which means each rectangle's height is determined by the function value at the right endpoint of the subinterval.
Let's break this down:

• The subintervals identified provide the base width of the rectangles, which is \( \Delta x = \frac{\pi}{2} \).
• The right endpoints, calculated earlier, provide the points where we evaluate the function to get the height of each rectangle.
• For this function f(x) and interval, we calculated three right endpoints: \( \pi \), \( \frac{3\pi}{2} \), and \( 2\pi \).
• The Riemann sum here is calculated by summing the area of each rectangle: \( A \approx \sum_{i=1}^{3} f(x_i) \cdot \Delta x \).

In essence, the final summed area \( 3\pi \) provides the approximate area under the curve from the start to the end of the interval using right endpoints.

Function Evaluation

Function evaluation is crucial in determining the height of rectangles in a Riemann sum approximation.
In this problem, we examine \( f(x) = 2 + \sin(2x) \), and we calculate it at each of the right endpoints.
Here's how it's done:

• For \( x = \pi \), \( f(\pi) = 2 + \sin(2\pi) = 2 + 0 = 2 \).
• For \( x = \frac{3\pi}{2} \), \( f\left(\frac{3\pi}{2}\right) = 2 + \sin(3\pi) = 2 + 0 = 2 \).
• For \( x = 2\pi \), \( f(2\pi) = 2 + \sin(4\pi) = 2 + 0 = 2 \).

These values tell us the height of each rectangle at these points.
This consistent functional evaluation at the endpoints allows us to calculate the Riemann sum precisely, ensuring our approximation of the area is as accurate as possible. In this case, every function evaluation resulted in a height of 2, indicating that each rectangle contributes equally to the sum.