Problem 1 - 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. $$ \int_{1}^{3} \frac{1}{x^{2}} d x $$

Step-by-Step Solution

Answer

The exact value of the integral is \(\frac{2}{3}\).

Step 1: Left Riemann Sum

To approximate the integral using the Left Riemann Sum, divide the interval \([1, 3]\) into 8 equal subintervals. The width of each subinterval is \( \Delta x = \frac{3 - 1}{8} = \frac{1}{4} \). The left endpoints of these subintervals are \(1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2}, \frac{11}{4} \). The Left Riemann sum is then:\[L_n = \sum_{i=0}^{7} f(x_i) \Delta x = \sum_{i=0}^{7} \frac{1}{x_i^2} \cdot \frac{1}{4}\]

Step 2: Right Riemann Sum

For the Right Riemann Sum, divide the interval \([1, 3]\) into 8 equal subintervals, and use the right endpoints, which are \(\frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2}, \frac{11}{4}, 3 \). The width of each subinterval is \( \Delta x = \frac{1}{4} \). The Right Riemann sum is given by:\[R_n = \sum_{i=1}^{8} f(x_i) \Delta x = \sum_{i=1}^{8} \frac{1}{x_i^2} \cdot \frac{1}{4}\]

Step 3: Trapezoidal Rule

Using the Trapezoidal Rule, approximate the integral with the same division of 8 subintervals. The formula is given by:\[T \approx \frac{\Delta x}{2} \left(f(x_0) + 2\sum_{i=1}^{7} f(x_i) + f(x_8)\right)\]where \( \Delta x = \frac{1}{4} \). Calculate the function's values at all endpoints and intermediate points same as in the Left and Right Riemann Sums.

Step 4: Parabolic Rule (Simpson's Rule)

To use the Parabolic Rule or Simpson's Rule with \(n=8\), ensure \(n\) is even (which it is). The formula is:\[S \approx \frac{\Delta x}{3} \left(f(x_0) + 4\sum_{i=1,3,5,7} f(x_i) + 2\sum_{i=2,4,6} f(x_i) + f(x_8)\right)\]Substitute \( \Delta x = \frac{1}{4} \) and calculate using the same endpoints as in the previous steps.

Step 5: Exact Value using the Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus states that if \( F(x) \) is an antiderivative of \( f(x) \), then:\[\int_{a}^{b} f(x) \, dx = F(b) - F(a)\]Find an antiderivative \( F(x) = -\frac{1}{x} \) for the integrand \( \frac{1}{x^2} \). Evaluate the result as:\[F(3) - F(1) = \left(-\frac{1}{3}\right) - \left(-1\right) = \frac{2}{3}\]

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to estimate the value of a definite integral. In essence, it approximates the region under a curve as a series of trapezoids, rather than rectangles like in Riemann sums. This alternative provides a better approximation, especially when the function is relatively smooth. The formula is: \[ T \approx \frac{\Delta x}{2} \left(f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n)\right) \] where \( \Delta x \) is the width of each subinterval and \( f(x) \) are the function values at endpoints and intermediate points.

• Each trapezoid spans one subinterval, with its bases being the values of the function at the respective endpoints.
• This method provides a decent approximation by averaging the heights of the function at the two ends of each subinterval.

The approximation improves as the number of subintervals increases and is particularly efficient when the function itself closely resembles straight lines within the subintervals.

Parabolic Rule (Simpson's Rule)

The Parabolic Rule, commonly known as Simpson's Rule, offers an even better approximation than the Trapezoidal Rule by fitting parabolas to segments of the curve you want to integrate. This method requires the number of intervals \( n \) to be even. The formula is: \[ S \approx \frac{\Delta x}{3} \left(f(x_0) + 4\sum_{i=1,3,5,\ldots}^{n-1} f(x_i) + 2\sum_{i=2,4,6,\ldots}^{n-2} f(x_i) + f(x_n)\right) \]

• Here, the multiplicative coefficients (4 and 2) reflect the weighting of contributions of the odd and even indexed points, capturing the behavior of the quadratic approximation.
• By using parabolas, this rule assumes the function behaves somewhat quadratically over each pair of intervals, which typically results in more accurate predictions, especially with smooth functions.

Simpson's Rule is particularly powerful in reducing the error due to its parabolic approximation, making it a popular choice when the function is relatively smooth over the interval of integration.

Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus is a profound result that connects the concepts of differentiation and integration, showing how one can obtain the exact value of a definite integral given an antiderivative. Essentially, if \( F(x) \) is an antiderivative of \( f(x) \), then the theorem states: \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]

• This principle reveals that the accumulation of a function's area is precisely equivalent to subtracting the values of its antiderivative evaluated at the upper limit and the lower limit of integration.
• It provides a straightforward method for calculating exact integrals, grounding the process of reversing differentiation to find net area or accumulated value.

Applying this theorem, as seen in the step-by-step solution, uncovers the strong link between integration and antiderivatives, simplifying the process of computing exact areas under curves.

Antiderivative

An antiderivative of a function \( f(x) \) is another function \( F(x) \) whose derivative is \( f(x) \). In simpler terms, it's a reverse operation to differentiation.

• Finding the antiderivative is essential in evaluating definite integrals using the Second Fundamental Theorem of Calculus.
• The notation for the antiderivative of \( f(x) \) is often represented as \[ F(x) = \int f(x) \, dx \]

In context with the example provided, for the function \( \frac{1}{x^2} \), an antiderivative \( F(x) \) is found to be \(-\frac{1}{x}\). Understanding antiderivatives helps in tackling key calculus problems, especially those related to area under a curve or total accumulation, and shines a light on how differentiation links back to integration.