Problem 62 - Calculus Early Transcendentals

Question

Complete the following steps for the given integral and the given value of \(n\) a. Sketch the graph of the integrand on the interval of integration. b. Calculate \(\Delta x\) and the grid points \(x_{0}, x_{1}, \ldots, x_{n},\) assuming a regular partition. c. Calculate the left and right Riemann sums for the given value of \(n\). d. Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral. $$\int_{-1}^{1} \pi \cos \left(\frac{\pi x}{2}\right) d x$$

Step-by-Step Solution

Answer

Answer: The left Riemann sum underestimates the value of the definite integral, and the right Riemann sum overestimates the value of the definite integral.

Step 1: Sketch the graph of the integrand

On the interval of integration \([-1, 1]\), the function is $$f(x) = \pi \cos \left(\frac{\pi x}{2}\right)$$ To sketch the graph, observe that the function is continuous and oscillates between positive and negative values. It has a maximum value at \(x = 0\) and reaches zero at the endpoints of the interval \(x=-1\) and \(x=1\).

Step 2: Calculate the partition and grid points

We want to divide the interval \([-1, 1]\) into \(n\) equal partitions. The width of each partition is given by $$\Delta x = \frac{b-a}{n} = \frac{1-(-1)}{n} = \frac{2}{n} $$ The grid points are given by \(x_i = a + i\Delta x\), for \(i = 0, 1, \ldots, n\). So we have $$x_i = -1 + i \frac{2}{n}, \quad i = 0, 1, \ldots, n$$

Step 3: Calculate the left and right Riemann sums

The left Riemann sum is given by $$L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x = \sum_{i=0}^{n-1} \pi \cos\left(\frac{\pi (-1 + i(2/n))}{2}\right) \cdot \frac{2}{n} $$ The right Riemann sum is given by $$R_n = \sum_{i=1}^{n} f(x_i) \Delta x = \sum_{i=1}^{n} \pi \cos\left(\frac{\pi (-1 + i(2/n))}{2}\right) \cdot \frac{2}{n} $$

Step 4: Determine which Riemann sum underestimates and overestimates the value of the definite integral

Since the function is concave down on the interval \([-1, 0]\) and concave up on the interval \([0, 1]\), the left Riemann sum will underestimate the value of the definite integral on both subintervals, and therefore also underestimate the value of the definite integral on the whole interval \([-1, 1]\). Similarly, due to the concavity of the function on the interval \([-1, 1]\), the right Riemann sum will overestimate the value of the definite integral on both subintervals, and therefore also overestimate the value of the definite integral on the whole interval \([-1, 1]\). So, the left Riemann sum \(L_n\) underestimates the value of the definite integral, and the right Riemann sum \(R_n\) overestimates the value of the definite integral.

Key Concepts

Definite Integrals

A definite integral is an essential concept in calculus. It represents the signed area under a curve over a particular interval \([a, b]\). For a given function \(f(x)\), the notation \(\int_{a}^{b} f(x) \, dx\) signifies the definite integral from \(x = a\) to \(x = b\). Definite integrals have numerous applications, such as finding the total accumulated change of a function or the net area between a function and the x-axis over a specific interval.
In this exercise, the definite integral is \(\int_{-1}^{1} \pi \cos \left(\frac{\pi x}{2}\right) \, dx\). Here's a quick breakdown:

• Integrand: Function inside the integral \(\pi \cos \left(\frac{\pi x}{2}\right)\).
• Limits: Integration from \(-1\) to \(1\).
• Geometrically: Represents the net area between this trigonometric function and the x-axis, summarizing positive and negative areas.

Mastering definite integrals enables you to handle concrete mathematical problems like this one more effectively.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are crucial in analyzing periodic phenomena. In this exercise, the integrand involves the cosine function: \(\cos \left(\frac{\pi x}{2}\right)\).
It's transformed within the integral using a constant multiplier \(\pi\), showing that you can manipulate trigonometric functions to fit a specific scenario or problem.
Let’s break it down step by step:

• \(\textbf{Nature of Cosine:} \) Cosine is periodic and oscillates between –1 and 1.
• \(\textbf{Transformation:} \) \(\frac{\pi x}{2}\) modifies its standard period to adapt to the problem’s domain.
• \(\textbf{Graphical Understanding:} \) Knowing the standard wave-like graph structure of cosine helps in sketching it accurately, especially when transformations are applied.

These attributes underline the usefulness of trigonometric functions in representing oscillating behaviors like waves and vibrations within mathematics.

Concavity

Concavity describes the way a curve bends. If a function is curving upwards, it's concave up, and if it's bending downwards, it's concave down.
In our problem, the concavity of the function \(\pi \cos \left(\frac{\pi x}{2}\right)\) changes at different parts of the interval.

• For \([-1, 0]\): The function is concave down.
  This implies that the slope is decreasing — think of the top of a hill or crest.
• For \([0, 1]\): On this interval, it flips to concave up.
  Here, the slope starts increasing, like a valley’s upward path.

This changing nature of concavity influences the accuracy of left and right Riemann sums, helping us determine whether they underestimate or overestimate an integral as a result.

Graph Sketching

Graph sketching is a fundamental skill in visualizing math functions, providing insights that calculations alone might not convey. Sketching the function \(f(x) = \pi \cos \left(\frac{\pi x}{2}\right)\) over the interval \([-1, 1]\) involves a few mindful steps.
Begin by understanding cosine's characteristics, then apply transformations.

• \(\textbf{Key Points:} \) Mark points where the cosine function changes direction or meets the x-axis, like \(x = 0\), where it reaches a peak value \(\pi\), and \(x = \pm 1\), where it coincides with the axis.
• \(\textbf{Consider Oscillation:} \) Recognize its cyclical nature, evident through continuous wave patterns across the interval.
• \(\textbf{Plotting and Linking:} \) Connect marked points smoothly to form the characteristic wave shape, oscillating symmetrically over the interval.

This powerful visual tool aids in understanding both abstract and concrete mathematical concepts, especially when calculations become complex.