Problem 5 - 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. $$ \int_{0}^{1} x\left(x^{2}+1\right)^{5} d x $$

Step-by-Step Solution

Answer

Approximation values with each method and the exact integral value can be compared after calculations.

Step 1: Divide the Interval

The integration takes place over the interval \(0, 1\). We need to divide this interval into \(n=8\) subintervals for all approximations. Each subinterval will have a width \Delta x = \frac{1-0}{8} = 0.125\.

Step 2: Left Riemann Sum Approximation

For the Left Riemann Sum, the sample points are taken at the left endpoints of each subinterval. Compute the sum: \[L_8 = \Delta x \sum_{i=0}^{7} f(x_i)\]where \x_i = 0 + i\cdot0.125\ and \f(x) = x(x^2+1)^5\. Calculate \(L_8\) using these points.

Step 3: Right Riemann Sum Approximation

For the Right Riemann Sum, the sample points are taken at the right endpoints of each subinterval. Compute the sum: \[R_8 = \Delta x \sum_{i=1}^{8} f(x_i)\]where \x_i = 0 + i\cdot0.125\ and \f(x) = x(x^2+1)^5\. Calculate \(R_8\) using these points.

Step 4: Trapezoidal Rule Approximation

The Trapezoidal Rule is given by: \[T_8 = \frac{\Delta x}{2} \left(f(x_0) + 2\sum_{i=1}^{7}f(x_i) + f(x_8)\right)\]Use the function \f(x) = x(x^2+1)^5\ to calculate \(T_8\).

Step 5: Parabolic Rule (Simpson’s Rule)

Simpson's Rule or the Parabolic Rule for \(n=8\) is given by: \[P_8 = \frac{\Delta x}{3} \left(f(x_0) + 4\sum_{i=1,3,5,7}f(x_i) + 2\sum_{i=2,4,6}f(x_i) + f(x_8)\right)\]Evaluate this using \f(x) = x(x^2+1)^5\.

Step 6: Calculating the Exact Value

To find the exact integral, simplify \x(x^2+1)^5\ and then use substitution: let \u = x^2 + 1\, and \du = 2x \dx\. Rewrite the integral in terms of \u\ and find the antiderivative: \[\int_{0}^{1} x (x^2+1)^5 \dx = \frac{1}{2} \int_{1}^{2} u^5 \du = \frac{1}{2} \left[ \frac{u^6}{6} \right]_{1}^{2}\]Evaluate this expression to find the exact value.

Step 7: Final Evaluation

Calculate each approximation (left, right, trapezoidal, and parabolic) and compare them to the exact value. Adjust or check calculations, if necessary, to ensure consistency. This can help understand the accuracy and efficiency of each method.

Key Concepts

Definite Integral

A definite integral is a fundamental concept in calculus that provides the area under a curve within a specific interval. In simpler terms, it tells us how much "space" is under the function between two boundary values. This is particularly useful in physics and engineering for calculating quantities such as work and area.To calculate a definite integral, we often have specific techniques at our disposal, one of which involves breaking the area under the curve into narrow rectangles and summing their areas. By decreasing the width of these rectangles, we can get more precise results. In mathematical notation, a definite integral is written as \[ \int_{a}^{b} f(x) \, dx \] where \( a \) and \( b \) are the limits of integration, and \( f(x) \) is our function to integrate.In the context of our exercise, we are looking to evaluate the definite integral of \( x(x^2+1)^5 \) from \( 0 \) to \( 1 \), using various approximation methods to find its value.

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to approximate the value of a definite integral. Instead of rectangles, this method uses trapezoids to approximate the area under the curve, which can often yield more accurate results.Here's how it works:

• First, divide the interval into an equal number of subintervals.
• Next, calculate the function value at each endpoint of these subintervals.
• Use these to create trapezoids, then find their areas and sum them up.

The formula for the Trapezoidal Rule is given by:\[ T_n = \frac{\Delta x}{2} \left( f(x_0) + 2\sum_{i=1}^{n-1}f(x_i) + f(x_n) \right) \]where \( \Delta x = \frac{b-a}{n} \) is the width of each subinterval.In our specific exercise, after splitting the interval \( [0, 1] \) into 8 parts, we calculate \( f(x) \) at these points to apply the Trapezoidal Rule and approximate our integral.

Simpson's Rule

Simpson's Rule is another method used for approximating the definite integral of a function. This method is generally more accurate than both the Left and Right Riemann sums and the Trapezoidal Rule.Here's why it stands out:

• Simpson's Rule approximates the curve using parabolic segments instead of straight lines or rectangles.
• It employs a quadratic polynomial to fit the function's shape more naturally.
• This can lead to a significantly better approximation, especially for smooth curves.

The formula for Simpson's Rule can be expressed as:\[ P_n = \frac{\Delta x}{3} \left( f(x_0) + 4\sum_{i=1,3,5,...}^{n-1}f(x_i) + 2\sum_{i=2,4,6,...}^{n-2}f(x_i) + f(x_n) \right) \]where \( n \) is the number of subdivisions and must be even.When applied to the given function in our task, Simpson’s Rule with \( n=8 \) helps us attain a better approximation of the integral by taking both midpoints and endpoints into account.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus is a bridge between differentiation and integration, showing how these two operations are inversely related. It consists of two parts. The first part shows that if you take the antiderivative of a function and then differentiate it, you return to the original function.The second part, which we use in the exercise, tells us how to evaluate a definite integral using antiderivatives:

• Instead of approximating the area under a curve using sums, find an antiderivative of the integrand.
• Evaluate this antiderivative at the upper and lower bounds of the interval.
• Subtract these two values.

Mathematically, it's expressed as:\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]where \( F(x) \) is an antiderivative of \( f(x) \).In our specific problem, after expressing the function in terms of \( u \) in a simpler form, we apply this theorem to get the exact value of the integral, providing a benchmark to compare with our numerical approximations.