Area Calculation: Curve Xy=4, X-axis, X=1, X=3

Hey guys! Today, we're diving into a classic calculus problem: finding the area of a region bounded by a curve, the x-axis, and a couple of vertical lines. Specifically, we're going to tackle the curve xy = 4, the x-axis, and the lines x = 1 and x = 3. Buckle up, because we're about to embark on a mathematical adventure!

Understanding the Problem

Before we jump into calculations, let's make sure we understand what the problem is asking. We're given a curve defined by the equation xy = 4. This is a hyperbola. We're also given the x-axis (which is simply the line y = 0) and two vertical lines, x = 1 and x = 3. These lines define the left and right boundaries of the region we're interested in. Our goal is to find the area of the region enclosed by these four boundaries: the curve, the x-axis, and the two vertical lines.

Essentially, we're looking for the definite integral of the function defined by the curve between the limits x = 1 and x = 3. This is a fundamental concept in calculus, and mastering it will help you solve a wide variety of problems. The ability to visualize the region we're trying to find the area of is very important.

Visualizing the Region

It always helps to visualize what we're working with. Imagine the graph of xy = 4. It's a hyperbola that exists in the first and third quadrants. Since we're only considering the region between x = 1 and x = 3, we can focus on the part of the hyperbola in the first quadrant. The x-axis forms the lower boundary, and the lines x = 1 and x = 3 are vertical walls on either side. The region we're interested in is the area trapped between the curve, the x-axis, and these two lines.

Setting Up the Integral

The key to finding this area lies in integration. Remember that the definite integral of a function f(x) from a to b gives us the area under the curve f(x) between the lines x = a and x = b. In our case, a = 1 and b = 3. But what is our f(x)?

We have the equation xy = 4. To express y as a function of x, we simply solve for y: y = 4/x. Therefore, f(x) = 4/x. Now we can set up our definite integral:

Area = ∫[from 1 to 3] (4/x) dx

Why Integration Works

Think of integration as summing up an infinite number of infinitesimally thin rectangles. Each rectangle has a width of dx (an infinitely small change in x) and a height of f(x) (the value of the function at that x). The area of each rectangle is f(x) dx. By adding up the areas of all these tiny rectangles between x = 1 and x = 3, we get the total area under the curve.

Evaluating the Integral

Now that we've set up the integral, it's time to evaluate it. The integral of 1/x is ln|x| (the natural logarithm of the absolute value of x). Since we're dealing with positive values of x between 1 and 3, we can simply use ln(x). Therefore, the integral of 4/x is 4 ln(x).

So, we have:

Area = 4 ln(x) evaluated from 1 to 3

To evaluate this, we plug in the upper limit (3) and subtract the result of plugging in the lower limit (1):

Area = 4 ln(3) - 4 ln(1)

Remember that ln(1) = 0. So, the expression simplifies to:

Area = 4 ln(3) - 4 * 0 = 4 ln(3)

The Answer

Therefore, the area of the region bounded by the curve xy = 4, the x-axis, and the lines x = 1 and x = 3 is 4 ln(3). Looking back at the multiple-choice options, the correct answer is (b).

The final answer is (b) $4 \ln 3$

Key Takeaways

• Understanding the Problem: Always start by visualizing the region you're trying to find the area of.
• Setting Up the Integral: Express the curve as a function of x (i.e., y = f(x)) and determine the limits of integration.
• Evaluating the Integral: Use your knowledge of integration rules to find the definite integral. Don't forget to evaluate the integral at the upper and lower limits and subtract the results.
• Checking Your Answer: If possible, estimate the area visually to see if your answer makes sense.

Expanding Your Knowledge: Applications of Area Calculation

The ability to calculate the area under a curve isn't just a theoretical exercise. It has numerous applications in various fields, including:

Physics

In physics, the area under a velocity-time graph represents the displacement of an object. Similarly, the area under a force-displacement graph represents the work done by the force. These concepts are fundamental to understanding motion and energy.

For example, if you have a graph showing the velocity of a car over time, the area under the curve between two points in time tells you how far the car traveled during that time interval. This is a powerful tool for analyzing motion.

Engineering

Engineers use area calculations in various applications, such as determining the cross-sectional area of structural beams, calculating fluid flow rates, and designing aerodynamic surfaces. The principles of calculus are essential for ensuring the safety and efficiency of engineered systems.

Consider designing a bridge: engineers need to calculate the area of the bridge's supports to ensure they can withstand the weight and stress they will bear. Accurate area calculations are crucial for preventing structural failures.

Economics

In economics, the area under a demand curve can represent consumer surplus, while the area under a supply curve can represent producer surplus. These concepts are used to analyze market efficiency and welfare.

For example, the area between the demand curve and the price line represents the benefit consumers receive from purchasing a good or service at a price lower than what they would be willing to pay. This is a key concept in welfare economics.

Probability and Statistics

In probability and statistics, the area under a probability density function (PDF) represents the probability of an event occurring within a certain range. This is a fundamental concept in statistical inference and hypothesis testing.

Imagine a bell curve representing the distribution of heights in a population. The area under the curve between two height values represents the probability that a randomly selected person from that population will have a height within that range. This is a powerful tool for making predictions and inferences about populations.

Advanced Techniques and Considerations

While the basic principle of finding the area under a curve is straightforward, there are some advanced techniques and considerations to keep in mind for more complex problems:

Areas Between Curves

If you need to find the area between two curves, f(x) and g(x), you simply integrate the difference between the two functions: Area = ∫[from a to b] |f(x) - g(x)| dx. The absolute value ensures that you're always integrating a positive difference.

Improper Integrals

If the limits of integration are infinite or if the function has a discontinuity within the interval of integration, you'll need to use improper integrals. These require special techniques to evaluate.

Numerical Integration

For functions that are difficult or impossible to integrate analytically, you can use numerical integration techniques, such as the trapezoidal rule or Simpson's rule, to approximate the area.

Coordinate Systems

Sometimes, it's easier to calculate areas using different coordinate systems, such as polar coordinates. This is particularly useful for regions with circular or radial symmetry.

Conclusion

So there you have it! We've successfully navigated the process of finding the area bounded by a curve, the x-axis, and two vertical lines. Remember, the key is to understand the problem, set up the integral correctly, and evaluate it carefully. With practice, you'll become a pro at solving these types of problems. And remember, this skill isn't just for math class – it has real-world applications in physics, engineering, economics, and more.

Keep practicing, keep exploring, and most importantly, keep having fun with math!