Problem 21 - Calculus

Question

Draw \(y=1 / x^{2}\) for \(0

Step-by-Step Solution

Answer

Two rectangles' area is \(\frac{5}{2}\); four rectangles' area is approximately \(5.69\). The limit is infinity.

Step 1: Understanding the Function

The given function is \(y = \frac{1}{x^2}\). This is a hyperbola with a vertical asymptote at \(x = 0\), and it approaches infinity as \(x\) approaches zero from the right. You need to graph the function for the domain \(0 < x < 1\).

Step 2: Dividing the Interval and Drawing Rectangles

We divide the interval \((0, 1)\) into two equal parts: \( \left(0, \frac{1}{2} \right) \) and \( \left(\frac{1}{2}, 1 \right)\). For each part, draw a rectangle with the base of \( \frac{1}{2} \). Use the left endpoint \( x = \frac{1}{2} \) and the right endpoint \( x = 1 \) for the heights of the rectangles.

Step 3: Calculating the Area of Two Rectangles

For the first rectangle (from \(x=0\) to \(x=\frac{1}{2}\)), the height is \( \frac{1}{x^2} \bigg|_{x=\frac{1}{2}} \), which is \( \frac{1}{(\frac{1}{2})^2} = 4 \). The area is \(4 \times \frac{1}{2} = 2\). For the second rectangle (from \(x=\frac{1}{2}\) to \(x=1\)), the height is \( \frac{1}{1^2} = 1 \). The area is \(1 \times \frac{1}{2} = \frac{1}{2}\). Sum the areas: \(2 + \frac{1}{2} = \frac{5}{2}\).

Step 4: Dividing the Interval for Four Rectangles

Divide the interval \( (0, 1) \) into four equal parts: \( \left(0, \frac{1}{4} \right), \left( \frac{1}{4}, \frac{1}{2} \right), \left( \frac{1}{2}, \frac{3}{4} \right), \left( \frac{3}{4}, 1 \right) \). Each rectangle will now have a base of \(\frac{1}{4}\). Determine the height using the left endpoints of these intervals.

Step 5: Calculating the Area of Four Rectangles

For the rectangles, use heights obtained from \(x = \frac{1}{4}\) to \(x = 1\): - First rectangle: Height \( = \frac{1}{(\frac{1}{4})^2} = 16\), Area \( = 16 \times \frac{1}{4} = 4\).- Second rectangle: Height \( = \frac{1}{(\frac{1}{2})^2} = 4\), Area \( = 4 \times \frac{1}{4} = 1\).- Third rectangle: Height \( = \frac{1}{(\frac{3}{4})^2} = \frac{16}{9}\), Area \( = \frac{16}{9} \times \frac{1}{4} \approx 0.44\).- Fourth rectangle: Height \( = 1\), Area \( = 1 \times \frac{1}{4} = \frac{1}{4}\).Sum the areas: \(4 + 1 + 0.44 + \frac{1}{4} \approx 5.69\).

Step 6: Guessing the Limit

As you increase the number of rectangles, the sum of their area approximates the area under the curve. Since \(y = \frac{1}{x^2}\) doesn't have an upper bound, the limit of the area as the number of rectangles approaches infinity is infinity.

Key Concepts

Area Under Curve

When calculating the area under a curve, you seek to find the region between the curve and the x-axis. This region is significant in mathematics since it represents the integral of a function across a specific interval.
This concept becomes especially interesting with functions that do not easily yield a simple geometric area, requiring methods like Riemann sums to approximate these areas.

• The area under a curve can represent various things such as total distance, accumulated value, and more, depending on the context.
• To find this area accurately, integration is often employed. But in situations where integration isn't practical or possible, numerical approximation methods like the rectangles method are useful.

The rectangles method offers a simple way to estimate by dividing the area into small sections (rectangles) and summing their areas. As more rectangles are used, the approximation gets closer to the true area.

Hyperbola

Understanding a hyperbola is essential to new learners of calculus, especially when dealing with functions like \(y = \frac{1}{x^2}\). The hyperbola in this instance is a curve that never touches the axes but endlessly approaches them. It has a unique set of behaviors and characteristics:

• Hyperbolas have two branches. However, in the specified domain \((0 < x < 1)\), we're observing just one branch that moves along the positive side.
• As \(x\) approaches zero from the right, the curve climbs upwards steeply, indicating an increase in \(y\) values towards infinity. Conversely, as \(x\) heads towards 1, \(y\) smoothly decreases to smaller values.

This kind of behavior can make calculating the area under such a hyperbola either analytically or practically challenging, as the outputs diverge significantly based on the x-values chosen.

Infinite Limit

The concept of an infinite limit is crucial when analyzing functions that exhibit explosive growth or decay, similar to \(y = \frac{1}{x^2}\).
As you examine the function in the range \(0 < x < 1\), particularly close to \(x = 0\), the value of \(y\) becomes exceedingly large. This is an example of an infinite limit where the values trend without bound either towards positive or negative infinity:

• An infinite limit expresses that as \(x\) approaches a particular value, the function's output tends towards infinity.
• In this context, when estimating area using rectangles, it signifies that the area under the curve can keep increasing without bound as rectangles get finer and more numerous.

This explains why the area, when calculated with more rectangles, provides larger approximate values leading to the realization that it tends towards an infinite sum. It's a pivotal consideration to handle theoretical scenarios where boundaries of estimation are tricky to define.

Rectangles Method

The rectangles method, also known as a Riemann sum, is a vital tool for approximating the area under a curve. By breaking a domain into equal-sized sections and stacking rectangles under the curve, you derive an approximation for the total area:

• Each rectangle is typically characterized by its base and height. The base comes from how the domain is divided, and the height from the function value at a chosen point (often the left or right endpoint, or the midpoint).
• The area of each individual rectangle is calculated as the "base \( \times \) height" and, through summation, provides the estimated total area.

Increasing the number of rectangles usually leads to a better approximation because they fit more snugly under the curve. In exercises like those involving \(y = \frac{1}{x^2}\), this approach helps visualize and understand how the approximation improves as the rectangles become narrower. As you use more rectangles, you can see the sum of these rectangles' areas converging toward a solution or, in some cases, showing a behavior such as diverging towards infinity when the function grows unboundedly.