Problem 48 - Essential Calculus

Question

etermine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(x=\sin \theta,\) then \(\int_{-1}^{1} x^{2} \sqrt{1-x^{2}} d x=2 \int_{0}^{\pi / 2} \sin ^{2} \theta \cos ^{2} \theta d \theta\)

Step-by-Step Solution

Answer

The statement is false, because after substituting the function and simplifying, the equation obtained does not match the original equation in the problem statement. The reason is that the term \(\cos^2 (\theta)\) from the original statement is not present in the simplified equation, which has \(\cos (\theta)\) instead.

Step 1: Variable Substitution

Substitute \(x = \sin \theta\) into the integral \(\int_{-1}^{1} x^{2} \sqrt{1-x^{2}} dx\), to obtain: \(\int_{-\pi/2}^{\pi/2} \sin^2 \theta \cdot \cos \theta d\theta\). Note that the limits have changed because the values of \(x = \sin \theta\) range from \(-1\) to \(1\) when \(\theta\) varies from \(-\pi/2\) to \(\pi/2\).

Step 2: Simplify the Equation

Simplify the equation, which is the integral from \(-\pi/2\) to \(\pi/2\) of \(\sin^2 \theta \cdot \cos \theta d \theta\). Here we double the integral from 0 to \(\pi/2\) because the function \(\sin^2 \theta \cdot \cos \theta\) is symmetric around the y-axis, resulting in: \(2\int_{0}^{\pi / 2} \sin ^{2} \theta \cos \theta d \theta\).

Step 3: Compare both Sides

Now, it can be noticed that the simplified equation obtained from the substitution is not identical to the equation given in the problem statement, \((2\int_{0}^{\pi / 2} \sin ^{2} \theta \cos ^{2} \theta d \theta)\). The original equation has \(\cos^2 \theta\), whereas the equation obtained through substitution has \(\cos \theta\).

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify the integration of expressions involving square roots. The idea is to use trigonometric identities to transform a complex function into a simpler one. This is often useful when dealing with integrals that include the form \(\sqrt{1-x^2}\), which appears frequently in problems involving circles or ellipses.
In the given exercise, we use the substitution \(x = \sin \theta\) because it transforms the square root expression \(\sqrt{1-x^2}\) into \(\cos \theta\), based on the trigonometric identity \(\sin^2 \theta + \cos^2 \theta = 1\).

• This substitution changes the bounds of integration from \(-1\) to \(1\) for \(x\) to \(-\pi/2\) to \(\pi/2\) for \(\theta\), since \(\sin \theta\) covers the range \([-1, 1]\) in this interval.
• Trigonometric substitution helps simplify complex-looking integrals into more manageable trigonometric integrals that can be solved using standard techniques.

Definite Integrals

Definite integrals are a crucial part of calculus; they compute the accumulation of quantities, which could represent areas, probabilities, or other sums. When evaluating definite integrals using substitution, adjustments to the limits are crucial. These new limits ensure that the variable change correctly reflects the original bounds of the problem.
In this example, the transformation from \(x\) to \(\theta\) through \(x = \sin \theta\) modifies the integral's bounds from \(-1\) to \(1\) to \(-\pi/2\) to \(\pi/2\).

• The definite integral was initially from \(-1\) to \(1\), a typical range when involving functions like \(\sin \theta\).
• Correct limits are vital as they ensure the substituted integral matches the original problem within the defined space.

This process involves modifying the formula inside the integral to match these new boundaries, aligning it with transformations introduced by substitution.

Symmetry in Functions

Symmetry plays a significant role in simplifying definite integrals, especially with functions that are symmetric around an axis. If a function exhibits symmetry, it can reduce the complexity of solving integrals by allowing us to consider only part of the range.
In the given problem, we harness the symmetry of \(\sin^2 \theta \cdot \cos \theta\) about the y-axis. This functionality allows us to simplify the integral from \(-\pi/2\) to \(\pi/2\) into a doubled integral from \(0\) to \(\pi/2\).

• This is possible because the function \(\sin^2 \theta \cdot \cos \theta\) is an even function, meaning its graph looks the same when flipped over the y-axis.
• The use of symmetry quickly reduces computation since calculating a smaller range and doubling the result often involves less work.

By recognizing these symmetries, complex integrals become more manageable, turning difficult problems into straightforward calculations.