Problem 66 - Calculus of a Single Variable

Question

Using Symmetry The area of the region bounded by the graphs of \(y=x^{3}\) and \(y=x\) cannot be found by the single integral \(\int_{-1}^{1}\left(x^{3}-x\right) d x .\) Explain why this is so. Use symmetry to write a single integral that does represent the area.

Step-by-Step Solution

Answer

The difference between signed and actual area leads to the problem. To solve this, instead of using the integral \(\int_{-1}^{1}\left(x^{3}-x\right) d x\), the correct integral which uses symmetry to represent the total area is \(2 \int_{0}^{1}\left(x-x^{3}\right) d x\).

Step 1: Understanding the difference between signed and actual area

Signed area is defined by the integral and it can be positive or negative depending on the graph’s position relative to the x-axis. But the 'actual' area between two curves cannot be negative, which means we need to find the absolute value area.

Step 2: Plot the functions

To visualize, you can plot the functions \(y = x^3\) and \(y = x\) on the range \([-1, 1]\). The two graphs intersect at \(x = -1\), \(x = 0\), and \(x = 1\), forming two symmetrical regions about the y-axis.

Step 3: Define new integral using symmetry

To get the actual area, we need to integrate the absolute difference of the two functions, which can be represented by two pieces in this case. However, because of symmetry, the area of these two regions is the same, we only need to calculate the area of one piece(phases of curves from 0 to 1) and then double this area. Hence, the correct integral for this problem is \(2 \int_{0}^{1}\left|x^{3}-x\right| d x\) = \(2 \int_{0}^{1}\left|x-x^{3}\right| d x\), because \(x > x^3\) for \(x\) in the interval \((0, 1)\).

Step 4: Calculation

To calculate the integral, firstly map the absolute value function to \(x - x^3\) (Because \(x > x^3\)), and then calculate the definite integral: \(2 \int_{0}^{1}\left(x-x^{3}\right) d x\).

Key Concepts

Signed versus Actual Area

When studying calculus, particularly integration, the terms 'signed area' and 'actual area' frequently come into play when discussing areas between curves. Signed area refers to the result of a definite integral, where the area above the x-axis counts as positive and the area below counts as negative. This can be seen through the integral ewline \[\int_{a}^{b}(f(x) - g(x)) \text{d}x\]
where the function represented by \(f(x)\)is above, and \(g(x)\) lies below on the interval \([a, b]\).

However, when calculating the 'actual area' between two functions, we are interested in the aggregate of the region, irrespective of whether it lies above or below the x-axis. Negative results are not logical when we talk about physical space, which is why in certain cases, we must modify our integral to account for actual area. One common method is by taking the absolute value of the integrand, essentially flipping the negative areas above the x-axis.

In the provided example with the functions \(y = x^3\) and \(y = x\), the integral originally posed\(\int_{-1}^{1}(x^3 - x) \text{d}x\)could yield a signed area, which does not correctly represent the actual area we seek. To overcome this, we find the absolute difference between the functions, thus assuring a non-negative integrand and an accurate representation of the actual area.

Symmetry in Integration

The concept of symmetry in integration is a powerful tool for simplifying the calculation of areas between curves, especially when dealing with functions that are symmetrical across an axis. By recognizing symmetrical properties, we can often reduce the computational work needed.

In many cases, if a region bounded by symmetrical curves is symmetrical about the y-axis, as is with \(y = x^3\) and \(y = x\),we can exploit this symmetry. For the interval from \([-1, 1]\), the area on one side of the y-axis is a mirror image of the other. Hence, instead of computing the entire area in one go, we calculate the area on one side (like from 0 to 1) and then double it to account for the other symmetrical half.

In our exercise, after plotting the functions, we can observe this kind of symmetry. Consequently, we only need to address the interval from 0 to 1 to get half the area, then multiply by 2, which leads us to the corrected integral representation: \(2 \int_{0}^{1}|x^3 - x| \text{d}x\).By using symmetry, we achieve a simpler, more efficient calculation for the entire area between the curves.

Definite Integrals

Definite integrals play a crucial role in calculus as they allow us to calculate the accumulation of quantities, such as areas under curves. A definite integral has upper and lower limits, which are the bounds of integration, and it gives a real number that represents the total accumulation of the integrand between these limits.

Mathematically, a definite integral is expressed as \[\int_{a}^{b} f(x) \text{d}x\]and it represents the signed area under the curve \(f(x)\)from \(x = a\)to \(x = b\).The process to find a definite integral involves finding the antiderivative (also called the indefinite integral) of the function, and then applying the Fundamental Theorem of Calculus, which entails evaluating the antiderivative at the upper and lower bounds and subtracting the two.

For instance, in the step-by-step solution to our exercise problem, we are instructed to calculate a definite integral from 0 to 1 for the functions presented. This calculation involves finding the integral of the absolute difference between \(x^3\) and \(x\), and this integration process gives us the actual area under these curves for that specific interval. It's a clear demonstration of how definite integrals not only have theoretical importance but also practical applications, such as determining the space bounded between curves in the presented question.