Problem 4 - 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. $$ \int_{1}^{3} x \sqrt{x^{2}+1} d x $$

Step-by-Step Solution

Answer

The exact integral value is approximately 8.472.

Step 1: Divide the Interval

We start by dividing the interval \([1, 3]\) into 8 equal subintervals because we need to use Riemann sums and numerical methods. The width of each subinterval \( \Delta x\) is calculated as follows:\[\Delta x = \frac{3 - 1}{8} = 0.25.\]

Step 2: Calculate the Left Riemann Sum

For the left Riemann sum, use the left endpoints of each subinterval: \(1, 1.25, 1.5, 1.75, 2, 2.25, 2.5, 2.75\). The left Riemann sum is given by:\[L = \sum_{i=0}^{7} f(x_i) \times \Delta x = \sum_{i=0}^{7} x_i \sqrt{x_i^2+1} \times 0.25.\]Compute the sum, substituting each value of \(x_i\).

Step 3: Calculate the Right Riemann Sum

For the right Riemann sum, use the right endpoints: \(1.25, 1.5, 1.75, 2, 2.25, 2.5, 2.75, 3\). The right Riemann sum is:\[R = \sum_{i=1}^{8} f(x_i) \times \Delta x = \sum_{i=1}^{8} x_i \sqrt{x_i^2+1} \times 0.25.\]Compute the sum with these endpoints.

Step 4: Apply the Trapezoidal Rule

In the trapezoidal rule, the average of the heights at the endpoints gives the area of each trapezoid. The trapezoidal approximation is:\[T = \frac{\Delta x}{2} \Big( f(x_0) + 2\sum_{i=1}^{7} f(x_i) + f(x_8) \Big).\]Calculate this using the same subdivision points.

Step 5: Apply the Parabolic (Simpson's) Rule

Simpson's Rule requires an even number of subintervals; since \(n=8\), we can apply it. The formula is:\[S = \frac{\Delta x}{3} \Big( f(x_0) + 4\sum_{i=1,3,5,7} f(x_i) + 2\sum_{i=2,4,6} f(x_i) + f(x_8) \Big).\]Compute this value using the specified points.

Step 6: Use the Fundamental Theorem of Calculus

To find the exact value, first find the antiderivative of \(f(x) = x\sqrt{x^2+1}\). Let \(u = x^2 + 1\), then \(du = 2x \cdot dx\), hence:\[\int x\sqrt{x^2+1} \, dx = \frac{1}{2} \int u^{1/2} \, du = \frac{1}{2} \cdot \frac{2}{3} u^{3/2} + C = \frac{1}{3} (x^2 + 1)^{3/2} + C.\]Evaluate it from 1 to 3:\[F(3) - F(1) = \left[ \frac{1}{3} (3^2 + 1)^{3/2} \right] - \left[ \frac{1}{3} (1^2 + 1)^{3/2} \right].\]Simplify this to find the exact integral value.

Key Concepts

Riemann Sums

Riemann sums are a fundamental concept in calculus for approximating the area under a curve. They break a continuous interval into smaller subintervals to sum up slices of area. Each slice is essentially a rectangle, and the rectangles approximate the curve's area. There are different types of Riemann sums, primarily left Riemann sum and right Riemann sum, determined by the point from which the function's value is taken in the subinterval.

In a left Riemann sum, the left endpoint of each subinterval is used. For our function \( f(x) = x\sqrt{x^2+1} \) over the interval [1, 3], the subintervals are defined with common difference \( \Delta x=0.25 \), and the function value at each left endpoint is taken to express the rectangles' heights. Similarly, a right Riemann sum uses the right endpoint of each subinterval.

The formula for a Riemann sum is expressed as\[ L = \sum_{i=0}^{n-1} f(x_i) \Delta x \] for the left sum and \[ R = \sum_{i=1}^{n} f(x_i) \Delta x \] for the right sum, where \( x_i \) are the respective endpoints.

Trapezoidal Rule

The trapezoidal rule enhances the basic concept of the Riemann sum by using trapezoids instead of rectangles to approximate areas under a curve. Each trapezoid's top edge is a line that connects the endpoints of consecutive subintervals. This results in a more accurate approximation, particularly for curves with straight-line sections.

For the function \( f(x) = x \sqrt{x^2+1} \) on the interval [1, 3], divide it into 8 equal segments with a width \( \Delta x = 0.25 \). The area of each trapezoid can be calculated by averaging the values at the endpoints of each subinterval. The full approximation is then given by:

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

This formula effectively considers both the function's heights at the start and end, and the doubled sum of heights within the interval.

Simpson's Rule

Simpson's Rule is a powerful technique for numerical integration, providing a very accurate approximation by using parabolic arches rather than straight lines or trapezoids. For this approach, divide the integration interval into an even number of subintervals, as it relies on fitting these quadratics.

For the function in question, the interval is divided into 8 equal parts, which is suitable for Simpson's Rule:

• \( S = \frac{\Delta x}{3} \left( f(x_0) + 4 \sum_{\text{odd } i} f(x_i) + 2 \sum_{\text{even } i} f(x_i) + f(x_n) \right) \)

This formula balances the approximation by giving more weight (i.e., multiplying by 4) to the odd-indexed sums of function values, enhancing accuracy for quadratic models. The alternation of coefficients breaks the bounded intervals into parabolic arcs that fit the curvature of the original function well.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concept of an antiderivative with a definite integral, providing a formal method to compute the exact area under a curve. This theorem states that if a function is continuous over an interval, then integration and differentiation are inverse processes.

In practical terms, finding an antiderivative of the function \( f(x) \), allows one to determine the definite integral over a specified range. For the given function \( f(x) = x \sqrt{x^2+1} \), an antiderivative is needed:

• Firstly, apply substitution where \( u = x^2 + 1 \) and \( du = 2x \, dx \).
• This leads to \( \int x\sqrt{x^2+1} \, dx = \frac{1}{3} (x^2+1)^{3/2} \).

To find the definite integral from 1 to 3, evaluate \((x^2 + 1)^{3/2}\) at the bounds and subtract: \( F(3) - F(1) \). This gives the exact value, filling in any gaps left by approximation methods.