Question

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

Short answer

The exact value is \( \frac{2}{3} \). Approximate values will differ slightly.

Step-by-step solution

Divide the Interval

The interval of integration is from 1 to 3. For all methods, divide this interval into 8 subintervals (since \(n = 8\)). The width of each subinterval, \(\Delta x\), is computed as \(\Delta x = \frac{3-1}{8} = \frac{1}{4}\).

Left Riemann Sum Calculation

For the left Riemann sum, evaluate the integrand at the left endpoint of each subinterval:\[L_8 = \sum_{i=0}^{7} f(x_i) \cdot \Delta x = \left( f(1) + f\left(1.25\right) + \ldots + f(2.75)\right) \cdot \frac{1}{4}\]where \(f(x) = \frac{1}{x^2}\).Calculate each term and sum them.

Right Riemann Sum Calculation

For the right Riemann sum, evaluate the integrand at the right endpoint of each subinterval:\[R_8 = \sum_{i=1}^{8} f(x_i) \cdot \Delta x = \left( f(1.25) + f(1.5) + \ldots + f(3)\right) \cdot \frac{1}{4}\]Evaluate each term similarly to step 2.

Trapezoidal Rule Calculation

The Trapezoidal Rule can be expressed as:\[T_8 = \frac{\Delta x}{2} \left( f(1) + 2f(1.25) + 2f(1.5) + \ldots + 2f(2.75) + f(3) \right)\]Compute this using the same function values as in the Riemann sum calculations.

Parabolic Rule (Simpson's Rule) Calculation

For the Parabolic Rule, which is equivalent to Simpson's Rule, compute:\[S_8 = \frac{\Delta x}{3} \left( f(1) + 4f(1.25) + 2f(1.5) + 4f(1.75) + \ldots + 4f(2.75) + f(3) \right)\]This involves computing terms with varying coefficients 4 and 2, ending and starting with coefficient 1.

Exact Solution Using the Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus states that:\[\int_{a}^{b} f(x) \, dx = F(b) - F(a)\]Where \( F'(x) = f(x) \). Here, find the antiderivative\[F(x) = -\frac{1}{x}\]Thus,\[\int_{1}^{3} \frac{1}{x^2} \, dx = -\frac{1}{3} + \frac{1}{1} = \frac{2}{3}\]

Compare Approximate and Exact Solutions

Calculate each approximate value using the formulas above from Step 2 to Step 5, and compare these to the exact value \( \frac{2}{3} \). This shows the accuracy of each method.

Key concepts

Definite Integral

A definite integral is a way to calculate the area under a curve on a graph, or more precisely, the area between the curve defined by a function and the x-axis, from one point (a) to another point (b) on the axis. In the expression \( \int_{a}^{b} f(x) \, dx \), \( a \) and \( b \) are the limits of integration, and \( f(x) \) is the function from which the integral is being calculated.
The process of integration essentially sums an infinite number of infinitesimally small rectangles that fit under the curve.
In the exercise, the integral \( \int_{1}^{3} \frac{1}{x^2} \, dx \) is evaluated to find this net area between \( x = 1 \) and \( x = 3 \). The answer gives a specific numerical value that represents this area. Understanding definite integrals is crucial as they provide solutions in diverse fields such as physics and engineering.

Trapezoidal Rule

The Trapezoidal Rule is a numerical method for approximating the value of a definite integral. This method is especially useful when the antiderivative of a function is challenging or impossible to determine analytically.
It works by dividing the area under the curve into a series of trapezoids rather than rectangles. For each subinterval between points \( x_i \) and \( x_{i+1} \), a trapezoid is formed. The formula for the area of each trapezoid is given by \( T_n = \frac{\Delta x}{2} \left( f(x_0) + 2f(x_1) + \cdots + 2f(x_{n-1}) + f(x_n) \right) \), where \( \Delta x \) is the width of each subinterval.
In the given exercise, the Trapezoidal Rule with \( n=8 \) simplifies the process of finding the integral \( \int_{1}^{3} \frac{1}{x^2} \, dx \) by providing an approximation that gets closer to the exact integral value.

Simpson's Rule

Simpson's Rule is another technique for approximating a definite integral, providing more accuracy than the Trapezoidal Rule by using parabolic segments. Instead of straight lines or trapezoids, Simpson's Rule uses parabolic arcs to approximate the segments of curves.
Simpson's Rule is defined for an even number of intervals as \( S_n = \frac{\Delta x}{3} \left( f(x_0) + 4f(x_1) + 2f(x_2) + \cdots + 4f(x_{n-1}) + f(x_n) \right) \). The coefficients alternate between 4 and 2, indicating midpoints are emphasized more, briefly touching each endpoint once.
In the exercise, this method is referred to as the Parabolic Rule, and it's known for producing more accurate results for smooth functions. For the function \( \frac{1}{x^2} \), Simpson's Rule provides a nuanced approximation that is often closer to the exact integral compared to other methods.

Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus is a profound principle that connects differentiation and integration, two of calculus's central concepts. Simply put, if you have an antiderivative \( F(x) \) of a function \( f(x) \), then the definite integral of \( f \) from \( a \) to \( b \) can be directly computed as \( F(b) - F(a) \).
This theorem simplifies the calculation of definite integrals by leveraging the antiderivative, providing the exact accumulation of change from \( a \) to \( b \). For the integral \( \int_{1}^{3} \frac{1}{x^2} \, dx \), finding the antiderivative \( F(x) = -\frac{1}{x} \) allows us straightforwardly to replace and compute \( F(3) - F(1) \) to get an exact value of \( \frac{2}{3} \).
The power of this theorem lies in its ability to transform complex integration problems into simpler substitution tasks.