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Magazine Chapter 5: Problem 74
Verify that \(f(x)=\sin \sqrt[3]{x}\) is an odd function and use that fact to show that $$ 0 \leqslant \int_{-2}^{3} \sin \sqrt[3]{x} d x \leqslant 1 $$
Short Answer
Expert verified
\(f(x) = \sin \sqrt[3]{x}\) is an odd function; integration confirms \(0 \leq \int_{-2}^{3} \sin \sqrt[3]{x} \, dx \leq 1\).
Step by step solution
01
Check if \( f(x) \) is an odd function
A function \( f(x) \) is considered odd if \( f(-x) = -f(x) \). In our case \( f(x) = \sin \sqrt[3]{x} \), so we need to check if the function satisfies this condition. For \( f(-x) \), we have:\( f(-x) = \sin \sqrt[3]{-x} = \sin(-\sqrt[3]{x}) = -\sin \sqrt[3]{x} \).\Since \( f(-x) = -f(x) \), the function \( f(x)=\sin \sqrt[3]{x}\) is indeed an odd function.
02
Use the properties of odd functions to evaluate the integral
For any function \( f(x) \) that is odd, it can be shown that \( \int_{-a}^{a} f(x) \, dx = 0 \). However, our integral is not perfectly symmetrical around zero since it runs from \(-2\) to \(3\). We split it into two parts: \( \int_{-2}^{0} \sin \sqrt[3]{x} \, dx \) and \( \int_{0}^{3} \sin \sqrt[3]{x} \, dx \). The integral from \(-2\) to \(0\) will not necessarily be zero since it doesn't symmetrically cancel, but we can verify it will not substantially exceed certain values.
03
Estimate the integral between positive and negative limits
Since \( f(x) \) is odd, the contribution from \(-2\) to \(0\) and \(0\) to \(2\) cancel each other out: \(\int_{-2}^{2} \sin \sqrt[3]{x} \, dx = 0\). We are left with evaluating \( \int_{2}^{3} \sin \sqrt[3]{x} \, dx \), which has a positive result since \( \sin \) is positive in this domain. \(\sin \theta\) takes values between \(-1\) and \(1\), hence the integral's maximum possible contribution is \(1\) (though it actually will be less than 1 due to the specific bounds).
04
Final evaluation and comparison
To extend this to include \([0, 3]\), we only need to consider the positive integral part between \(2\) and \(3\). Generally, the function \( \sin \sqrt[3]{x} \) is quite small across this range for cubic roots of numbers close to 3, meaning the area under the curve is quite limited. With this understanding, we find that the full integration from \(-2\) to \(3\) results in a value more than \(0\) but less than \(1\), matching the inequality \(0 \leq \int_{-2}^{3} \sin \sqrt[3]{x} \, dx \leq 1\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Definite Integrals
Definite integrals allow us to calculate the net area under a curve within a particular interval. Think of it as finding how much of the function lies above and below the x-axis between two points. The notation \( \int_a^b f(x) \, dx \) expresses the integral from \( a \) to \( b \) of the function \( f(x) \).
It's like summing up infinitely many tiny rectangles whose height is given by \( f(x) \) and width is infinitesimally small. This calculation gives us the total signed area. That means areas above the x-axis count as positive and those below as negative.
In this particular problem, evaluating the integral \( \int_{-2}^{3} \sin \sqrt[3]{x} \, dx \) involves understanding how much positive and negative area is covered due to the nature of the sine function and its domain. Because sine can oscillate, definite integrals can sometimes be pleasingly simple or surprisingly tricky to evaluate by estimation or symmetry.
It's like summing up infinitely many tiny rectangles whose height is given by \( f(x) \) and width is infinitesimally small. This calculation gives us the total signed area. That means areas above the x-axis count as positive and those below as negative.
In this particular problem, evaluating the integral \( \int_{-2}^{3} \sin \sqrt[3]{x} \, dx \) involves understanding how much positive and negative area is covered due to the nature of the sine function and its domain. Because sine can oscillate, definite integrals can sometimes be pleasingly simple or surprisingly tricky to evaluate by estimation or symmetry.
Properties of Odd Functions
Odd functions have a unique property: they are symmetric about the origin. Mathematically, this symmetry is shown when a function \( f(x) \) fulfills the condition \( f(-x) = -f(x) \).
Visualize this as the graph of the function flipping over the origin. This property is essential when working with integrals because it can simplify calculations. If the limits of integration are symmetric around zero, the integral of an odd function over that symmetric interval will always be zero.
In our case, even though the interval \([-2, 3]\) is not symmetrical, the odd nature of \( \sin \sqrt[3]{x} \) allows us to consider how parts of the interval interact. Specifically, the sections of the integral from \(-2\) to \(0\) and from \(0\) to \(2\) cancel each other out, emphasizing the powerful property of odd functions in integral calculus.
Visualize this as the graph of the function flipping over the origin. This property is essential when working with integrals because it can simplify calculations. If the limits of integration are symmetric around zero, the integral of an odd function over that symmetric interval will always be zero.
In our case, even though the interval \([-2, 3]\) is not symmetrical, the odd nature of \( \sin \sqrt[3]{x} \) allows us to consider how parts of the interval interact. Specifically, the sections of the integral from \(-2\) to \(0\) and from \(0\) to \(2\) cancel each other out, emphasizing the powerful property of odd functions in integral calculus.
Trigonometric Functions
Trigonometric functions, such as the sine and cosine, are foundational in mathematics due to their periodic, oscillating nature. The sine function, \( \sin x \), is particularly significant in many calculus problems.
The sine function has a characteristic wave shape, repeating every \(2\pi\). It oscillates between -1 and 1, which means its integral over any symmetric interval centered at the origin would sum to zero if it was considered alone.
For the function \( \sin \sqrt[3]{x} \), the trigonometric aspect adds a layer of complexity. The cube root alters the usual interpretation of \(x\) in the sine function, adjusting how quickly it passes through its peaks and troughs. This means, between certain bounds, the value that \( \sin \sqrt[3]{x} \) can attain is constrained more sharply compared to a typical sine curve. Consequently, this plays a role in ensuring integrals, such as that from \(2\) to \(3\), contribute less to the final sum, confirming the problem's inequality.
The sine function has a characteristic wave shape, repeating every \(2\pi\). It oscillates between -1 and 1, which means its integral over any symmetric interval centered at the origin would sum to zero if it was considered alone.
For the function \( \sin \sqrt[3]{x} \), the trigonometric aspect adds a layer of complexity. The cube root alters the usual interpretation of \(x\) in the sine function, adjusting how quickly it passes through its peaks and troughs. This means, between certain bounds, the value that \( \sin \sqrt[3]{x} \) can attain is constrained more sharply compared to a typical sine curve. Consequently, this plays a role in ensuring integrals, such as that from \(2\) to \(3\), contribute less to the final sum, confirming the problem's inequality.
