Problem 46 - Calculus Volume 1

Question

[T] Use a computer algebra system to compute the Riemann sum, \(L_{N}, \quad\) for \(\quad N=10,30,50\) for \(f(x)=\sqrt{1-x^{2}}\) on [-1,1]

Step-by-Step Solution

Answer

As \( N \) increases from 10 to 50, the Riemann sum \( L_N \) approaches \( \pi/2 \).

Step 1: Understanding the Function and Interval

The function provided is \( f(x) = \sqrt{1-x^2} \) and the interval is \([-1, 1]\). We need to calculate the Riemann sum, which is an approximation of the integral of this function over the given interval using rectangles. \( L_N \) represents the Riemann sum using the left endpoints of \( N \) subintervals.

Step 2: Define the Subinterval Width

To compute the Riemann sum, we divide the interval \([-1,1]\) into \( N \) equal subintervals. The width \( \Delta x \) of each subinterval is given by:\[ \Delta x = \frac{b-a}{N} = \frac{1 - (-1)}{N} = \frac{2}{N}. \]

Step 3: Determine the Left Endpoints

For each subinterval, we determine the left endpoint. For a general subinterval \( i \), where \( i = 0, 1, 2, ..., N-1 \), the left endpoint is:\[ x_i = a + i\Delta x = -1 + i\left(\frac{2}{N}\right). \]

Step 4: Compute the Riemann Sum Formula

The Riemann sum \( L_N \) using the left endpoints is calculated using:\[ L_N = \sum_{i=0}^{N-1} f(x_i) \Delta x = \sum_{i=0}^{N-1} \sqrt{1 - (x_i)^2} \cdot \frac{2}{N}. \]

Step 5: Calculate for \( N = 10 \)

Set \( N = 10 \). Compute \( \Delta x = \frac{2}{10} = 0.2 \). The sum becomes:\[ L_{10} = \frac{2}{10} \sum_{i=0}^{9} \sqrt{1 - \left(-1 + i \cdot 0.2 \right)^2}. \] Evaluate this sum using a computer algebra system to get a numerical approximation.

Step 6: Calculate for \( N = 30 \)

Set \( N = 30 \). Compute \( \Delta x = \frac{2}{30} \approx 0.0667 \). The sum is given by:\[ L_{30} = \frac{2}{30} \sum_{i=0}^{29} \sqrt{1 - \left(-1 + i \cdot 0.0667 \right)^2}. \] Use a computer algebra system to compute the numerical approximation.

Step 7: Calculate for \( N = 50 \)

Set \( N = 50 \). Compute \( \Delta x = \frac{2}{50} = 0.04 \). The sum becomes:\[ L_{50} = \frac{2}{50} \sum_{i=0}^{49} \sqrt{1 - \left(-1 + i \cdot 0.04 \right)^2}. \] Use a computer algebra system to evaluate this sum.

Step 8: Interpretation of Results

The values of \( L_{10} \), \( L_{30} \), and \( L_{50} \) give increasingly accurate approximations of the area under \( f(x) = \sqrt{1-x^2} \) on \([-1,1]\). As \( N \) increases, we expect the Riemann sum to converge closer to the actual integral value, which represents the area of a semicircle with radius 1, equal to \( \pi/2 \).

Key Concepts

Computer Algebra System

A computer algebra system (CAS) is a type of software designed to perform symbolic mathematics. Unlike typical calculators, which work with numbers and approximate values, a CAS is capable of manipulating mathematical expressions in symbolic form. This makes it incredibly useful for complex tasks like solving integrals, algebraic manipulation, and, in our case, calculating Riemann sums.

When we use a CAS for computing Riemann sums, it helps automate the cumbersome numerical calculations involved. By defining the function, range, and number of subintervals, a CAS can quickly compute the sum of all rectangular areas, giving us a numerical approximation of the integral.

• Automates complicated calculations
• Ensures more accurate results
• Reduces computation time

Using a CAS in exercises involving integrals allows us to focus more on understanding the underlying concepts without bogging down due to arithmetic and repetitive steps.

Integral Approximation

Integral approximation is the process of estimating the value of an integral. For functions where finding an exact integral is difficult or impossible, we use approximation techniques like the Riemann sum.

The idea is to approximate the area under a curve by dividing it into simpler shapes, such as rectangles. The more subdivisions you make, the closer your approximation will be to the true value. With Riemann sums, as the number of intervals, denoted by \(N\), increases, the Riemann sum becomes more accurate.

• Offers a practical method to estimate integrals
• Effective when analytical integration isn't feasible
• Precision improves with more subdivisions

Understanding integral approximation is crucial for many fields like engineering and physics, where precise calculations are often required.

Left Endpoint Method

The left endpoint method is a specific technique used in calculating Riemann sums. Here, the left endpoint of each subinterval is used to determine the height of the rectangles used in the approximation.

For this method:

• Divide the whole interval into \(N\) equal parts.
• Use the starting point of each interval to calculate the rectangle's height.
• Sum the areas of all rectangles for the final approximation.

The left endpoint method is straightforward and often used because it's simple to implement. However, the choice of endpoint (left, right, or midpoint) can affect the accuracy. In cases where the function is decreasing, this method might underestimate the integral. But as \(N\) increases, the approximation will converge to the actual value, regardless of the endpoint chosen.

Semicircle Area

In the context of the given exercise, the function \(f(x) = \sqrt{1-x^2}\) represents the upper half of a circle with a radius of 1. This makes the task at hand an estimation of the area of a semicircle.

The integral of this function over \([-1, 1]\) gives the area of a semicircle, which is \(\frac{\pi}{2}\). The whole circle has an area of \(\pi r^2\), and since \(r = 1\), the circle's area is \(\pi\). The semicircle is exactly half of this, hence \(\frac{\pi}{2}\).

• The function describes a semicircle due to its form \(\sqrt{1-x^2}\).
• The integral represents the area under this curve on the defined interval.
• The exercise demonstrates how Riemann sums approximate this area.

Understanding semicircle areas is important because it connects geometry with calculus, providing a useful visual representation for solving integrals.