Problem 2 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-6, use the methods of (1) left Riemann sum, (2) right Riemann sum, (3) Trapezoidal Rule, (4) Parabolic Rule with \(n=8\) to approximate the definite integral. Then use the Second Fundamental Theorem of Calculus to find the exact value of each integral. $$ \int_{1}^{3} \frac{1}{x^{3}} d x $$

Step-by-Step Solution

Answer

The exact value of the integral is \( \frac{4}{9} \).

Step 1: Define the Function and Subdivide the Interval

The given integral is \( \int_{1}^{3} \frac{1}{x^{3}} \, dx \). We are asked to approximate this integral using Riemann sums and other rules with \( n = 8 \), which means we divide the interval \([1, 3]\) into 8 equal subintervals. The width of each subinterval is \( \Delta x = \frac{3-1}{8} = \frac{1}{4} \).

Step 2: Left Riemann Sum Approximation

For the left Riemann sum, we use the left endpoint of each subinterval to calculate the height of the rectangles. The left endpoints are \( x_0 = 1, x_1 = 1.25, \ldots, x_7 = 2.75 \). Calculate \( \text{LR} = \sum_{i=0}^{7} \frac{1}{x_i^3} \Delta x \).

Step 3: Right Riemann Sum Approximation

For the right Riemann sum, use the right endpoint of each subinterval. The right endpoints are \( x_1 = 1.25, x_2 = 1.5, \ldots, x_8 = 3 \). Calculate \( \text{RR} = \sum_{i=1}^{8} \frac{1}{x_i^3} \Delta x \).

Step 4: Trapezoidal Rule Approximation

The Trapezoidal Rule is an average of the left and right Riemann sums. Apply the formula: \( \text{Trapezoidal} = \frac{\Delta x}{2} \left( f(x_0) + 2 \sum_{i=1}^{7} f(x_i) + f(x_8) \right) \).

Step 5: Parabolic Rule (Simpson's Rule) Approximation

Use Simpson's Rule, which is a parabolic (quadratic) approximation. It requires an even number of subintervals. Apply the formula: \( \text{Simpson's} = \frac{\Delta x}{3} \left( f(x_0) + 4 \sum_{i\text{ odd}} f(x_i) + 2 \sum_{i\text{ even}} f(x_i) + f(x_n) \right) \).

Step 6: Exact Integral Using the Second Fundamental Theorem of Calculus

The integral \( \int_{1}^{3} \frac{1}{x^{3}} \, dx \) can be solved exactly using integration techniques. The antiderivative of \( \frac{1}{x^3} \) is \( -\frac{1}{2x^2} \). Evaluate \( F(x) = -\frac{1}{2x^2} \) at 3 and 1: \( F(3) - F(1) = -\frac{1}{18} + \frac{1}{2} \). Calculate the exact value of the integral.

Key Concepts

Riemann Sums

Riemann Sums are a method for approximating the definite integral of a function, where we sum the areas of rectangles under a curve. Imagine carving the area under a curve into rectangles - the taller these rectangles, the more accurate our approximation becomes.

There are different types of Riemann Sums, including:

• Left Riemann Sum: Uses the left endpoint of subintervals to determine the height of rectangles. Although it can underestimate or overestimate the area, depending on the function's behavior, it provides an easy way to start approximating integrals.
• Right Riemann Sum: This sum uses the right endpoint for height calculations, and similarly, it may overestimate or underestimate the true area. The choice of left or right depends on the problem specifics. The approximation improves by increasing the number of subintervals which decreases their width.

Both methods involve taking the sum of function values at specified points multiplied by the width of each subinterval, denoted by \(\Delta x\). In practice, these methods can be quite close to the true integral.

Trapezoidal Rule

The Trapezoidal Rule refines the idea of Riemann sums by opting to approximate the region under a curve with trapezoids instead of rectangles.

This rule takes into account both the left and right endpoints of each interval to form each trapezoid, which generally gives a more accurate approximation than using rectangles alone.

Mathematically, it is expressed as:\[\text{Trapezoidal} = \frac{\Delta x}{2} \left( f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right)\]

This formula calculates the average of the left and right Riemann sums, effectively smoothing out their individual errors. It is especially useful in understanding areas when the function is unknown or hard to work with.

Simpson's Rule

Simpson's Rule is a more powerful technique that further improves upon the Trapezoidal Rule by using parabolic or quadratic approximations. It offers highly accurate results with fewer subintervals, making it a favorable choice in numerical integration.

This rule works by approximating the curve with parabolas passing through the values of the function at specific points.

The formula is:\[\text{Simpson's} = \frac{\Delta x}{3} \left( f(x_0) + 4 \sum_{i\text{ odd}} f(x_i) + 2 \sum_{i\text{ even}} f(x_i) + f(x_n) \right)\]

Simpson's Rule requires an even number of intervals. It works best when the function is smooth and continuous, fitting parabolas more smoothly than straight lines or trapezoids.

Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus is a profound concept that ties together the world of antiderivatives and definite integrals. It states that if \( f \) is continuous over an interval \([a, b]\) and \( F \) is an antiderivative of \( f \), then:\[\int_a^b f(x) \, dx = F(b) - F(a)\]

This means the definite integral of a function can be found using its antiderivative, significantly simplifying the calculation of areas under curves.

In the context of our problem, the antiderivative of \( \frac{1}{x^3} \) is \( -\frac{1}{2x^2} \). Applying the theorem, we evaluate this at our bounds 1 and 3, calculating the difference \( F(3) - F(1) \), providing the precise area, unlike approximations.