Problem 36 - Calculus of a Single Variable

Question

Finding the Area of a Region In Exercises \(31-36,\) (a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region analytically, and (c) use the integration capabilities of the graphing utility to verify your results. $$ f(x)=\frac{6 x}{x^{2}+1}, \quad y=0, \quad 0 \leq x \leq 3 $$

Step-by-Step Solution

Answer

The area bounded by the graphs of the functions \(f(x)=\frac{6x}{x^2+1}\) and \(y=0\) between \(x=0\) and \(x=3\) is derived through the calculation of a definite integral. This area can be verified using a graphing tool's definite integration feature.

Step 1: Graph the Functions

Use a graphing tool to plot the two functions \(f(x)=\frac{6x}{x^2+1}\) and \(y=0\) within the interval \(0 \leq x \leq 3\).

Step 2: Identifying the Bounded Area

The area to be computed is the one that is enclosed between the curve of the function \(f(x)\) and the \(x\)-axis within the interval \(0 \leq x \leq 3\).

Step 3: Compute Area Analytically

The area is calculated as the definite integral of the absolute value of the function across the interval, which is \( \int_{0}^{3} |\frac{6x}{x^2+1}| dx \). Since function is always positive in given range, the absolute value is not needed. Consequently, find the antiderivative of \(f(x)\), and evaluate it at the limits 0 and 3. The difference will yield the area underneath the curve of \(f(x)\) within the interval.

Step 4: Verification

Use the definite integration functionality available in graphing utilities to check the result derived from step 3 directly. Both results should be equal.

Key Concepts

Graphing Functions

Graphing functions involves plotting the given mathematical equations on a coordinate plane. In this exercise, we are interested in graphing the function \( f(x) = \frac{6x}{x^2 + 1} \) and comparing it to the line \( y = 0 \), which represents the x-axis. The purpose is to visualize the curves and identify the region bounded by these functions between \( x = 0 \) and \( x = 3 \).

When graphing, it is helpful to

• Input the function into a graphing utility like a calculator or online software.
• Ensure the graph covers the correct interval \( 0 \leq x \leq 3 \).
• Identify where the functions intersect or where they enclose an area, which in this case is above the x-axis and below \( f(x) \).

Graphing these functions provides a visual understanding of the space we will examine further through analytical methods.

Area Under Curve

The area under the curve is a fundamental concept in calculus for understanding the space between a curve and the x-axis. In the exercise, the area is calculated between the graph of \( f(x) = \frac{6x}{x^2 + 1} \) and the line \( y = 0 \) from \( x = 0 \) to \( x = 3 \).

This area represents the integral of the function over the given interval. Specifically,

• Use the definite integral \( \int_{0}^{3} \frac{6x}{x^2 + 1} \, dx \).
• By calculating this definite integral, we find how much space the curve covers above the x-axis across our specified range.

The area reflects the accumulation of small segments (or slices) stacked horizontally along the interval \( 0 \leq x \leq 3 \). Understanding this accumulation helps in interpreting many physical phenomena and mathematical applications.

Antiderivative

An antiderivative is a function whose derivative is the original function. Finding the antiderivative allows us to evaluate definite integrals, providing the area under a curve.

In this exercise, we need to calculate the antiderivative of \( f(x) = \frac{6x}{x^2 + 1} \).

• Use integration rules or lookup tables to find the antiderivative.
• For this specific function, appropriate substitutions or techniques like partial fraction decomposition might be necessary, depending on your familiarity with integration methods.
• The fundamental theorem of calculus tells us that we can find the definite integral by evaluating the antiderivative at the upper limit \( x = 3 \) and subtracting the value of the antiderivative at the lower limit \( x = 0 \).

The resulting computation gives us the precise area beneath the function and above the x-axis between \( x = 0 \) and \( x = 3 \).

Verification with Graphing Utilities

Verification with graphing utilities adds confidence to our analytical solutions. After calculating the area using calculus, it's wise to double-check the result with technology.

To verify using a graphing utility:

• Enter the function and set the integration feature on the interval \( 0 \leq x \leq 3 \).
• Observe the numerical output which represents the area under the function.
• Compare this output with your hand-computed result from the definite integral.

Matching results not only confirms the manual calculation but also reinforces the utility's capability in performing correct mathematical operations. Always ensure the tool is correctly set up to cover the intended interval for an accurate comparison.