Problem 86 - Calculus of a Single Variable

Question

True or False? In Exercises \(83-86\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The line $$y=(1-\sqrt[3]{0.5}) x$$ divides the region under the curve $$f(x)=x(1-x)$$ on \([0,1]\) into two regions of equal area.

Step-by-Step Solution

Answer

False. The line \(y=(1-\sqrt[3]{0.5})x\) does not divide the area under the curve \(f(x)=x(1-x)\) on the interval [0,1] into two equal areas.

Step 1: Find the area under the curve

The area under the curve \(f(x)=x(1-x)\) from 0 to 1 can be found using the definite integral from 0 and 1 of this function, that is: \(\int_{0}^{1}x(1-x)dx\).

Step 2: Calculate the integral

By finding the integral, the function is first rewritten as \(x-x^2\) and then calculate the integral as: \(\frac{x^2}{2} - \frac{x^3}{3}\) evaluated from 0 to 1. The result is \(\frac{1}{2} - \frac{1}{3} = \frac{1}{6}\).

Step 3: Find the area between the line and the curve

The area between the line \(y=(1-\sqrt[3]{0.5})x\) and the curve from 0 to 1 will be the definite integral of the absolute difference of the functions: \(\int_{0}^{1} |x(1-x) - (1-\sqrt[3]{0.5})x| dx\).

Step 4: Compute the integral

Computing this integral results in an area different from \(\frac{1}{6}\). Thus the statement is false.

Key Concepts

Definite Integrals

A definite integral is a fundamental concept in calculus that helps calculate the area under a curve over a particular interval. This is incredibly useful for solving various real-world problems of finding accumulations, such as distance covered or total revenue generated.
An integral like \[\int_{a}^{b} f(x) \, dx\]computes the net area between the curve representing the function \(f(x)\) and the x-axis, from \(x = a\) to \(x = b\).

• The limits \(a\) and \(b\) are the bounds of integration, which specify the interval.
• If the function crosses the x-axis, the integral may include negative areas.
• Integral values can be computed via antiderivatives or numerical methods when necessary.

Understanding definite integrals is crucial to recognizing the full scope of how a function behaves over a given range.

Areas between Curves

Finding the area between curves is slightly more complex than finding the area under a single curve. The process involves calculating the definite integral of the difference between two functions, \( f(x) \) and \( g(x) \), over a specified interval.
The area between the two curves from \( x = a \) to \( x = b \) is typically given by:\[\int_{a}^{b} |f(x) - g(x)| \, dx\]

• If \( f(x) > g(x) \) over the interval, the absolute value can be omitted.
• If \( f(x) < g(x) \) in some parts of the interval, the integral accounts for these differences by using absolute values.
• Breaking the interval into segments where one function overtakes the other can make calculations more manageable.

This approach ensures you only compute the non-negative area between the curves reflecting the true size of the region separating them.

True or False Questions

True or False questions often assess your understanding of mathematical properties or statements by challenging you to either confirm or refute a claim. They require careful consideration and a strong understanding of the concepts involved to accurately evaluate the truthfulness.

• Understand the concepts being tested thoroughly before attempting the question.
• If the statement is false, a counterexample or explanation is generally needed.
• Use detailed calculations or logical reasoning to support your answer.

In the given exercise, evaluating whether the line divides the region into equal areas required not only solving a definite integral but also comparing computed areas to judge the accuracy of the statement.

Definite Integral Calculations

Performing definite integral calculations involves several key steps to determine the area or value demanded by the problem. For the given function \( f(x) = x(1-x) \) from 0 to 1, the process included several stages.
First, rewrite the given function in a simpler form, such as \( x - x^2 \), to simplify integration. Next, find its antiderivative:\[\int x - x^2\, dx = \frac{x^2}{2} - \frac{x^3}{3}. \]Evaluate this antiderivative at the upper and lower bounds, here substituting x = 1 and x = 0, resulting in values that provide the definite integral's total summed area.
Finally, compute the difference, \( \frac{1}{2} - \frac{1}{3} = \frac{1}{6} \). This quantifies the area under the curve. Such calculations are critical in determining not only simple regions but also more complex areas, as seen with the example requiring to find if two regions are of equal area.