Problem 4 - Calculus

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

Step-by-Step Solution

Answer

Use each numerical method to approximate. The exact value is \( \frac{1}{3}(28) - \frac{1}{3}(2 \sqrt{2}) \).

Step 1: Left Riemann Sum

To approximate the integral using the left Riemann sum, divide the interval \([1, 3]\) into \(n = 8\) subintervals with equal width \( \Delta x = \frac{3-1}{8} = \frac{1}{4} \). The approximate value is \( \sum_{i=0}^{7} f(x_i) \Delta x \). Here, \( f(x) = x \sqrt{x^2 + 1} \) and \( x_i = 1 + i \cdot \frac{1}{4} \). Calculate each term and sum them up.

Step 2: Right Riemann Sum

For the right Riemann sum, the interval division is the same with \( \Delta x = \frac{1}{4} \), but use the right endpoints: \( x_i = 1 + (i+1) \cdot \frac{1}{4} \) for \( i = 0, 1, 2, \ldots, 7 \). Calculate \( \sum_{i=0}^{7} f(x_i) \Delta x \) using \( f(x) = x \sqrt{x^2 + 1} \).

Step 3: Trapezoidal Rule

With the trapezoidal rule, compute the approximation by evaluating \( \frac{\Delta x}{2} \left[ f(x_0) + 2 \sum_{i=1}^{7} f(x_i) + f(x_8) \right] \), with \( x_i = 1 + i \cdot \frac{1}{4} \). Calculate each term and their sum.

Step 4: Parabolic (Simpson's) Rule

Simpson's rule requires an even number of subintervals (\(n=8\)). Compute using \( \frac{\Delta x}{3} \left[ f(x_0) + 4 \sum_{i=1,3,5,7} f(x_i) + 2 \sum_{i=2,4,6} f(x_i) + f(x_8) \right] \). Evaluate for each term and coordinate the sum.

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

To find the exact value, first find an antiderivative of \( f(x) = x \sqrt{x^2 + 1} \). Using substitution, let \( u = x^2 + 1 \), then \( du = 2x \, dx \). The integral becomes \( \frac{1}{2}\int \sqrt{u} \, du \), evaluating which gives \( \frac{1}{3} (x^2 + 1)^{3/2} + C \). Evaluate at the boundaries: \([1,3]\), which yields: \( \left[ \frac{1}{3}(9+1)^{3/2} - \frac{1}{3}(1+1)^{3/2} \right] \).

Key Concepts

Riemann Sum

Riemann sums provide a way to approximate the definite integral of a function over an interval. There are two common types: left Riemann sum and right Riemann sum.

The left Riemann sum uses the left endpoints of each subinterval to calculate the sum. This involves dividing the interval \( [a, b] \) into \( n \) equal subintervals, finding the function's value at each left endpoint, and multiplying by the subinterval width \( \Delta x \).
For our problem, \( \Delta x = \frac{1}{4} \), and we calculate the sum: \( \sum_{i=0}^{7} f(x_i) \Delta x \).

The right Riemann sum, on the other hand, uses the right endpoints. This again involves dividing the interval into \( n \) subintervals, but instead, you find the function's value at each right endpoint: \( x_i = a + (i+1) \cdot \Delta x \).
Compute the sum: \( \sum_{i=1}^{8} f(x_i) \Delta x \).

These approximations give us a sense of the area under the curve by using rectangles, and while they may not be perfectly accurate, they provide a useful estimate.

Trapezoidal Rule

The trapezoidal rule is another method for approximating the value of a definite integral. It improves upon the accuracy of Riemann sums by using trapezoids instead of rectangles to estimate the area under a curve.

To apply the trapezoidal rule, divide the interval into \( n \) equal subintervals, similar to Riemann sums. However, you calculate the area of trapezoids that form between these subintervals. The formula for the trapezoidal rule is: \[ \frac{\Delta x}{2} \left[ f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right] \]
Here, \( f(x_0) \) and \( f(x_n) \) are the function values at the endpoints of the interval, and the other terms are double-weighted because they contribute to two trapezoids each.

This method often provides a better approximation than using Riemann sums as it accounts for the curvature of the function rather than assuming it is flat over each subinterval.

Simpson's Rule

Simpson's Rule offers an even more refined approximation for definite integrals by using parabolas instead of rectangles or trapezoids. It's more accurate when the function behaves well over the interval being considered.

Simpson's Rule requires the number of subintervals, \( n \), to be even, and it uses the following formula: \[ \frac{\Delta x}{3} \left[ f(x_0) + 4 \sum_{i=1,3,5,\ldots} f(x_i) + 2 \sum_{i=2,4,6,\ldots} f(x_i) + f(x_n) \right] \]
In this method, \( f(x_0) \) and \( f(x_n) \) are still single-weighted, but the coefficients of 4 and 2 alternate among the other function values. This unique weighting allows the formula to account more precisely for the shape of the function.

Simpson's Rule is particularly effective for functions that are well approximated by even simple curve shapes, like parabolas, and usually results in highly accurate approximations.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects the practice of differentiation with integration, providing an elegant way to compute exact values for definite integrals.

This theorem is composed of two parts. The second part, which is relevant here, states that if \( F \) is an antiderivative of \( f \) over an interval \( [a, b] \), then the definite integral of \( f \) from \( a \) to \( b \) is given by \[ F(b) - F(a) \].
An antiderivative is essentially a function whose derivative matches \( f(x) \).

In the exercise, we first find the antiderivative of \( f(x) = x \sqrt{x^2 + 1} \) by substitution or other methods. After obtaining the antiderivative \( F(x) \), we evaluate \( F(x) \) at the upper and lower limits 3 and 1 respectively, to find the exact integral's value.

This process highlights the power of calculus, transforming a complicated computational process into a simpler evaluation thanks to known antiderivatives.