Problem 4 - Calculus Early Transcendentals: Pearson New International Edition

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

Step-by-Step Solution

Answer

Left and right sums, trapezoidal and parabolic rules yield approximations; exact value is \( \frac{1}{3}(10 \sqrt{10} - 2 \sqrt{2}) \).

Step 1: Calculate abla x for Riemann Sums and Trapezoidal/Parabolic Rule

To begin, find the partition width \( \Delta x \). With \( a=1 \), \( b=3 \), and \( n=8 \): \[ \Delta x = \frac{b-a}{n} = \frac{3-1}{8} = \frac{1}{4} \].

Step 2: Left Riemann Sum

To calculate the left Riemann sum, evaluate the function at each left endpoint. Use the partition points: \( x_0 = 1, x_1=1.25, \ldots, x_7 = 2.75 \). The formula is:\[ L = \Delta x \sum_{i=0}^{n-1} f(x_i) = \frac{1}{4}\left[f(1) + f(1.25) + ... + f(2.75)\right] \]Calculate each \( f(x_i) \) and sum them up.

Step 3: Right Riemann Sum

Evaluate the function at each right endpoint using the points: \( x_1=1.25, x_2=1.5, \ldots, x_8=3 \). The formula is:\[ R = \Delta x \sum_{i=1}^{n} f(x_i) = \frac{1}{4}\left[f(1.25) + f(1.5) + ... + f(3)\right] \]Proceed to compute each \( f(x_i) \) and sum them.

Step 4: Trapezoidal Rule

The trapezoidal rule formula is:\[ T = \frac{\Delta x}{2} \left( f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) \]Use the points from \( x_0=1 \) to \( x_8=3 \). Calculate each \( f(x_i) \) as in previous steps.

Step 5: Parabolic (Simpson's) Rule

Use Simpson's rule, applicable since \( n \) is even:\[ S = \frac{\Delta x}{3} \left( f(x_0) + 4 \sum_{i=1,3,5,7} f(x_i) + 2\sum_{i=2,4,6} f(x_i) + f(x_8) \right) \]Calculate each \( f(x_i) \) for the points \( x_0=1, x_1=1.25, \ldots, x_8=3 \), and substitute them in.

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

Define an antiderivative for \( f(x) = x \sqrt{x^2 + 1} \). Let \( u = x^2 + 1 \), then \( du = 2x dx \). Thus, the integral becomes:\[ \int x \sqrt{x^2 + 1} \, dx = \int \frac{1}{2} \sqrt{u} \, du = \frac{1}{3} (x^2 + 1)^{3/2} \]Evaluate the definite integral using this antiderivative from 1 to 3:\[ \frac{1}{3} ((3^2 + 1)^{3/2} - (1^2 + 1)^{3/2}) = \frac{1}{3} (10 \sqrt{10} - 2\sqrt{2}) \].

Key Concepts

Riemann Sums

Riemann sums are a method for approximating the definite integral of a function. They're constructed by dividing the interval \(a, b\) into \(n\) subintervals of equal width \(\Delta x\), and calculating the sum of areas of rectangles.
These rectangles are either formed to the left or right of each subinterval, resulting in left or right Riemann sums respectively.
The left Riemann sum uses the left endpoints of subintervals as the height of the rectangles, giving the formula:

• \( L = \Delta x \sum_{i=0}^{n-1} f(x_i) \)

Conversely, the right Riemann sum employs right endpoints, resulting in:

• \( R = \Delta x \sum_{i=1}^{n} f(x_i) \)

Riemann sums provide a foundation for understanding integration concepts, offering insight into how areas under curves are numerically approximated. By increasing \(n\), the approximation becomes more accurate, closely estimating the true integral.

Trapezoidal Rule

The trapezoidal rule improves upon Riemann sums by approximating the area under a curve with trapezoids rather than rectangles.
This technique considers both endpoints of a subinterval, forming a trapezoid and more accurately estimating the area.
To use the trapezoidal rule, apply the following formula:

• \( T = \frac{\Delta x}{2} \left( f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n) \right) \)

Each term within the sum multiplies the function value at specified points, ensuring both endpoints of each trapezoid are adequately accounted for.
This rule often provides a better approximation than Riemann sums because trapezoids can better accommodate curve shapes.

Simpson's Rule

Simpson's Rule, also known as the Parabolic Rule, utilizes parabolas instead of straight lines to approximate the area under a curve.
This method uses a quadratic polynomial, resulting in a highly accurate approximation especially when the function is smooth and the interval is broken into an even number of subintervals.
Simpson's Rule employs the formula:

• \( S = \frac{\Delta x}{3} \left( f(x_0) + 4 \sum_{i=1,3,5,\ldots} f(x_i) + 2\sum_{i=2,4,6,\ldots} f(x_i) + f(x_n) \right) \)

The alternating coefficients of 4 and 2 are essential as they result from integrating a parabola, leading to higher precision than either the Riemann sums or Trapezoidal rule.

Second Fundamental Theorem of Calculus

The Second Fundamental Theorem of Calculus offers a profound connection between differentiation and integration.
It states that if \(F\) is an antiderivative of \(f\), then the definite integral from \(a\) to \(b\) of \(f\) is simply \(F(b) - F(a)\).
Mathematically, this is represented by:

• \( \int_a^b f(x) \, dx = F(b) - F(a) \)

This theorem provides an exact value of the definite integral rather than an approximation, as it uses antiderivatives to assess area under a curve directly.
For the exercise, it allows one to compute the precise area by evaluating the antiderivative of the given function at its upper and lower bounds.