Problem 32 - Thomas Calculus

Question

In Exercises \(29-32,\) graph each function \(f(x)\) over the given interval. Partition the interval into four subintervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum \(\Sigma_{k=1}^{4} f\left(c_{k}\right) \Delta x_{k},\) given that \(c_{k}\) is the (a) left-hand endpoint, (b) right- hand endpoint, (c) midpoint of the \(k\) th subinterval. (Make a separate sketch for each set of rectangles.) $$ f(x)=\sin x+1, \quad[-\pi, \pi] $$

Step-by-Step Solution

Answer

Graph \( f(x) = \sin x + 1 \) with three separate sets of rectangles for left-hand, right-hand, and midpoints over \([-\pi, \pi]\).

Step 1: Understand the Function and Interval

We are given the function \( f(x) = \sin x + 1 \) and need to analyze it over the interval \([-\pi, \pi]\). The function is a transformation of the sine function, shifted upward by 1 unit. We must partition this interval into four subintervals of equal length.

Step 2: Determine the Subintervals

Calculate the length of the entire interval: \( \pi - (-\pi) = 2\pi \). Dividing this interval into four equal parts, each subinterval has a length of \( \Delta x = \frac{2\pi}{4} = \frac{\pi}{2} \). The subintervals are: \([-\pi, -\frac{\pi}{2}], [-\frac{\pi}{2}, 0], [0, \frac{\pi}{2}], \text{and} [\frac{\pi}{2}, \pi]\).

Step 3: Calculate Left-Hand Endpoint Rectangles

For each subinterval, use the left-hand endpoint as \( c_k \) to find the height of each rectangle: - \( f(-\pi) = \sin(-\pi) + 1 = 0 + 1 = 1 \)- \( f(-\frac{\pi}{2}) = \sin(-\frac{\pi}{2}) + 1 = -1 + 1 = 0 \)- \( f(0) = \sin(0) + 1 = 0 + 1 = 1 \)- \( f(\frac{\pi}{2}) = \sin(\frac{\pi}{2}) + 1 = 1 + 1 = 2 \)Then, sketch rectangles with these heights within their respective subintervals.

Step 4: Calculate Right-Hand Endpoint Rectangles

For each subinterval, use the right-hand endpoint as \( c_k \) to find the height of each rectangle:- \( f(-\frac{\pi}{2}) = 0 \)- \( f(0) = 1 \)- \( f(\frac{\pi}{2}) = 2 \)- \( f(\pi) = \sin(\pi) + 1 = 0 + 1 = 1 \)Sketch the rectangles with these heights.

Step 5: Calculate Midpoint Rectangles

For each subinterval, use the midpoint as \(c_k\) to calculate the height:- Midpoint of \([-\pi, -\frac{\pi}{2}]\) is \(-\frac{3\pi}{4}: f(-\frac{3\pi}{4}) = \sin(-\frac{3\pi}{4}) + 1 = -\frac{\sqrt{2}}{2} + 1\)- Midpoint of \([-\frac{\pi}{2}, 0]\) is \(-\frac{\pi}{4}: f(-\frac{\pi}{4}) = \frac{1}{\sqrt{2}} + 1\)- Midpoint of \([0, \frac{\pi}{2}]\) is \(\frac{\pi}{4}: f(\frac{\pi}{4}) = \frac{1}{\sqrt{2}} + 1\)- Midpoint of \([\frac{\pi}{2}, \pi]\) is \(\frac{3\pi}{4}: f(\frac{3\pi}{4}) = -\frac{\sqrt{2}}{2} + 1\)Sketch rectangles using these heights.

Key Concepts

Graphing Functions

Understanding graphing functions is crucial for visualizing mathematical expressions. In this exercise, we are asked to graph the function \( f(x) = \sin x + 1 \) over the interval \([-\pi, \pi]\). This function represents a vertical shift of the classic sine wave. Instead of oscillating about the x-axis, it now oscillates one unit upward. The regular characteristics of the sine function, such as its periodicity and smooth wave-like appearance, remain, only shifted above the line \( y = 1 \).
To graph it correctly, note the key points: the peaks, troughs, and intercepts. The sine function reaches its peak value of \(1 + 1 = 2\) and its minimum of \(0 + 1 = 1\). As the sine function has a standard period of \(2\pi\), it completes one entire wave cycle over the given interval.

Subinterval Partitioning

Subinterval partitioning is essential for calculating Riemann sums. We divide the interval \([-\pi, \pi]\) into smaller sections to approximate the area under \( f(x)\). The first step is to calculate the interval's total length, which is \(2\pi\).

• Divide this length by the number of subintervals, which in this case is four. So, each subinterval has a length of \( \Delta x = \frac{2\pi}{4} = \frac{\pi}{2} \).
• List each subinterval: \([-\pi, -\frac{\pi}{2}], [-\frac{\pi}{2}, 0], [0, \frac{\pi}{2}], [\frac{\pi}{2}, \pi]\).

This partitioning allows us to calculate the Riemann sums using different methods, such as left-hand, right-hand, or midpoint approaches, by determining the appropriate value of \( c_k \) within each subinterval.

Sine Function Transformation

Transforming the sine function involves altering its amplitude, period, phase shift, or vertical shift. Here, in \( f(x) = \sin x + 1 \), the sine function is simply shifted upwards by 1 unit. This transformation affects the graph by relocating the midline. Instead of oscillating around the x-axis, the graph now oscillates about the line \( y = 1 \).

• The amplitude, which is the peak deviation from the midline, remains the same as the original sine function, \( 1 \).
• The period also remains \( 2\pi \), maintaining the classic sinusoidal cycle.

This vertical shift maintains the function’s shape but changes its baseline, affecting area calculations using Riemann sums.

Integration Approximation Techniques

Integration approximation techniques, like Riemann sums, are used to estimate the area under curves. These techniques involve summing up areas of geometric shapes, such as rectangles, that approximate the region under the function's graph. In the Riemann sum for \( f(x) = \sin x + 1 \), rectangles are drawn within each subinterval. You'll use different endpoints for calculating the rectangle heights:

• In the left-hand endpoint method, the height is determined by the function value at the left end of each subinterval.
• The right-hand endpoint uses the function value at the right end of each subinterval.
• The midpoint takes the function value at the middle of each subinterval. This method balances overestimates and underestimates better.

Each method gives a different approximation of the integral's value, demonstrating distinct estimation paradigms in integral calculus.