Problem 56 - 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. $$\int_{0}^{\pi / 2} \cos x d x ; n=4$$

Step-by-Step Solution

Answer

Answer: The Right Riemann Sum underestimates the value of the integral.

Step 1: Sketch the graph of the integrand

Draw the graph of the function \(y = \cos x\) on the interval \([0, \pi/2]\). The graph starts at \(\cos(0)=1\) and ends at \(\cos(\pi/2)=0\), decreasing monotonically.

Step 2: Calculate \(\Delta x\) and grid points

To find the width of the subintervals, we need to divide the entire interval by \(n=4\). Since the interval is \([0, \pi/2]\), the width of each subinterval is \(\Delta x = (\pi/2 - 0)/4 = \pi/8\). The grid points are given by \(x_i = 0+i\Delta x\), where \(i = 0, 1, 2, 3, 4\). In this case, we have: \(x_0 = 0\) \(x_1 = 0 + 1(\pi/8) = \pi/8\) \(x_2 = 0 + 2(\pi/8) = \pi/4\) \(x_3 = 0 + 3(\pi/8) = 3\pi/8\) \(x_4 = 0 + 4(\pi/8) = \pi/2\)

Step 3: Calculate left and right Riemann sums

To calculate the left Riemann sum, we evaluate the function at the left endpoint of each subinterval, and multiply by the width of the subinterval: Left Riemann Sum = \(\sum_{i=0}^{n-1} \cos(x_i)\Delta x = \cos(0)\frac{\pi}{8} + \cos(\frac{\pi}{8})\frac{\pi}{8} + \cos(\frac{\pi}{4})\frac{\pi}{8} + \cos(\frac{3\pi}{8})\frac{\pi}{8}\) To calculate the right Riemann sum, we evaluate the function at the right endpoint of each subinterval, and multiply by the width of the subinterval: Right Riemann Sum = \(\sum_{i=1}^{n} \cos(x_i)\Delta x = \cos(\frac{\pi}{8})\frac{\pi}{8} + \cos(\frac{\pi}{4})\frac{\pi}{8} + \cos(\frac{3\pi}{8})\frac{\pi}{8} + \cos(\frac{\pi}{2})\frac{\pi}{8}\)

Step 4: Determine underestimation/overestimation

As \(\cos x\) is a decreasing function over the interval \([0, \pi/2]\), the Left Riemann Sum will overestimate the value of the definite integral, while the Right Riemann Sum will underestimate the value of the definite integral.

Key Concepts

Integral Calculus

Integral calculus deals with the area under curves. It is a fundamental concept in calculus and helps us find the total accumulation of quantities, such as area, volume, and other sums over continuous intervals. The exercise in question involves understanding how to approximate the area under the curve of \(y = \cos x\) between \[0, \pi/2\]\, using Riemann sums.
When we compute an integral, we are essentially adding up infinitesimally small slices of area together to get the total area under a curve. This concept helps in determining values like distances, areas, and volumes, which don't necessarily have straightforward computations.
In the given exercise, the function \(\cos x\) signifies the graph whose area under the curve is being calculated. The integral \(\int_0^{\pi/2} \cos x \, dx\) represents the total area between the curve \(y = \cos x\), the \(x\)-axis, and the vertical lines \(x=0\) and \(x=\frac{\pi}{2}\).
This integration gives us the exact answer, but here we explore approximation methods like Riemann sums to build a better intuition of the integral calculus.

Trapezoidal Approximation

Trapezoidal approximation is one of the numerical methods to approximate the value of definite integrals, providing a balance between overestimating and underestimating compared to left and right Riemann sums.
The trapezoidal rule approximates the area under the curve by dividing the area into trapezoids rather than rectangles. For each section between two consecutive points on the function, you calculate the area of trapezoids instead of rectangles.
We calculate the sum of these trapezoids' areas to estimate the integral. The formula for a single trapezoid between points \(x_i\) and \(x_{i+1}\) is:

• Area of trapezoid = \(\frac{1}{2} (y_i + y_{i+1}) \Delta x \)

Here, \(y_i\) and \(y_{i+1}\) are the function values at \(x_i\) and \(x_{i+1}\), respectively, and \(\Delta x\) is the width of the corresponding interval.
The trapezoidal rule is often more accurate than basic Riemann sums because it accounts for the curve's slant by averaging the endpoints, leading to a closer approximation of the area under the curve, especially with a smaller \(\Delta x\).

Definite Integrals

Definite integrals represent the exact accumulation of a quantity, like area, under a curve over a specific interval. For the exercise, the definite integral \(\int_0^{\pi/2} \cos x \, dx\) shows how to accumulate the values of \(\cos x\) from \(x = 0\) to \(x = \frac{\pi}{2}\).
The definite integral has a clear meaning: it is the total area between the curve and the \(x\)-axis within the specified limits. The area calculated by integrating provides the exact answer compared to numerical methods, which give approximations.
For functions like \(\cos x\), which are continuous and smooth, definite integrals are straightforward to solve analytically using antiderivatives. In this case, the antiderivative of \(\cos x\) is \(\sin x + C\), and evaluating from \(0\) to \(\pi/2\) gives:

• \(\int_0^{\pi/2} \cos x \, dx = \sin\left(\frac{\pi}{2}\right) - \sin(0) = 1 - 0 = 1\)

This confirms the exact area under the curve between the limits.

Numerical Integration

Numerical integration involves calculating the integral of a function using approximation methods rather than analytical solutions. It is particularly useful when the function is complex, or its antiderivative is difficult to determine.
There are several methods for numerical integration, including Riemann sums, trapezoidal rule, and Simpson's rule, among others. In this exercise, both left and right Riemann sums are employed as a means to approximate the integral of \(\cos x\) over \([0, \pi/2]\).
Riemann sums work by splitting the integral into small, manageable intervals (or rectangles) and adding the area of these intervals. This method can be done with:

• The left Riemann sum, evaluating at the left endpoint of each subinterval.
• The right Riemann sum, evaluating at the right endpoint of each subinterval.

While these methods offer approximations, refining them with more intervals (a larger \(n\)) can yield results closer to the exact integral value. Numerical integration is essential in calculations where precision is crucial, and the function's complexity prevents easy integration.