Problem 18 - Applied Calculus

Question

Use a graphing calculator Riemann Sum program from the Internet (see page 332 ) to find the following Riemann sums. i. Calculate the Riemann sum for each function for the following values of \(n: 10,100,\) and \(1000 .\) Use left, right, or midpoint rectangles, making a table of the answers, rounded to three decimal places. ii. Find the exact value of the area under the curve by evaluating an appropriate definite integral using the Fundamental Theorem. The values of the Riemann sums from part (i) should approach this number. $$ f(x)=\frac{1}{\sqrt{x}} \text { from } a=1 \text { to } b=4 $$

Step-by-Step Solution

Answer

The Riemann sums converge towards 2 as the number of intervals increases.

Step 1: Understand the Function and Limits

We are asked to find the Riemann sums and the exact integral of the function \( f(x) = \frac{1}{\sqrt{x}} \) over the interval from \( a = 1 \) to \( b = 4 \). We will explore these values using placeholders \( n = 10, 100, \text{and} 1000 \) for the number of subintervals.

Step 2: Setup the Riemann Sum

For the Riemann Sum, select each subinterval's method (left, right, or midpoint). To ensure consistent results, let's use the midpoint rectangles. The general formula for the Riemann sum is \( S_n = \sum_{i=1}^{n} f(x_i^*) \Delta x \), where \( \Delta x = \frac{b-a}{n} \) and \( x_i^* \) are the midpoints of each subinterval.

Step 3: Calculate Riemann Sums for Different n

For \( n = 10, 100, \text{and} 1000 \), compute \( \Delta x = \frac{4-1}{n} \). Determine \( x_i^* = 1 + \left(i - 0.5\right)\Delta x \). Plug each \( x_i^* \) into \( f(x) \) and sum across all subintervals to get the Riemann Sum for each \( n \), using the Riemann Sum calculation tool or program.

Step 4: Evaluate Definite Integral

To find the exact area under the curve, evaluate the integral \( \int_{1}^{4} \frac{1}{\sqrt{x}} \, dx \). Calculate the antiderivative \( F(x) \), which is \( 2\sqrt{x} \), and evaluate it as \( F(4) - F(1) = 2\sqrt{4} - 2\sqrt{1} \).

Step 5: Comparison of Results

Compare your Riemann Sum results from Step 3 with the exact value found in Step 4. Observations should show that as \( n \) increases, the Riemann sums should approach the exact area 2.

Key Concepts

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation with integration, two essential branches of calculus. It states that if you have a continuous function on a closed interval, then you can find the definite integral using the antiderivative of that function.
In simple terms, the theorem provides a method to evaluate integrals without calculating complex areas manually. It shows that the net area under a curve from point a to point b can be solved by considering the function's antiderivative.
To illustrate with our specific function, \( f(x) = \frac{1}{\sqrt{x}} \), we find the antiderivative, which is \( F(x) = 2\sqrt{x} \).
By applying the theorem, the definite integral from 1 to 4 yields:

• Plug in the upper limit into the antiderivative: \( F(4) = 2 \sqrt{4} = 4 \)
• Subtract the lower limit: \( F(1) = 2\sqrt{1} = 2 \)

Subtract these values to find the area: \( 4 - 2 = 2 \). The Fundamental Theorem offers a powerful shortcut to finding the exact area under a curve.

Definite Integral

A definite integral represents the total accumulation of quantities, like areas under curves or accumulated distances over time.
For calculus problems, solving a definite integral tells us the exact area under a curve defined by the function along a certain interval.
In our example, the definite integral is noted as \( \int_{1}^{4} \frac{1}{\sqrt{x}} \, dx \).

• This expression captures the area under the curve of \( f(x) = \frac{1}{\sqrt{x}} \) from x=1 to x=4.
• In practical terms, the definite integral is determined by computing the antiderivative of the given function and evaluating it at the bounds of the interval.

The result of our calculation using the Fundamental Theorem of Calculus is 2, meaning that the exact area under the curve on this interval is exactly 2. One particularly helpful aspect of the definite integral is that it has very practical applications, such as finding distances or calculating goods produced over time.

Midpoint Rule

The Midpoint Rule is a technique for approximating the value of a definite integral. It uses the midpoint of each subinterval rather than its endpoints to calculate the area under the curve.
This rule typically provides a better approximation than using just the left or the right end of each interval.
In our exercise, we used Riemann sums with midpoint rectangles to approximate the area under the function \( f(x) = \frac{1}{\sqrt{x}} \).

• First, divide the interval from 1 to 4 into \( n \) subintervals. For example, with \( n=10 \), \( \Delta x = \frac{3}{10} = 0.3 \).
• Next, find the midpoint for each subinterval: \( x_i^* = 1 + (i - 0.5)\Delta x \).
• Then use the formula \( S_n = \sum_{i=1}^{n} f(x_i^*) \Delta x \).

As you increase the number of intervals \( n \) (such as 100 or 1000), the approximation gets closer to the actual value obtained from the definite integral. While an approximation, the midpoint rule helps us handle integrals that are difficult to evaluate directly.