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Chapter 2: Problem 47

By noting that for \(0 \leq \eta \leq 1, \eta^{1 / 2} \geq \eta^{3 / 4} \geq \eta\), prove that $$ \frac{2}{3} \leq \frac{1}{a^{5 / 2}} \int_{0}^{a}\left(a^{2}-x^{2}\right)^{3 / 4} d x \leq \frac{\pi}{4} $$

Short Answer

Expert verified
Prove via integral transformation substituting limits, proving correct bounds on 蟺/4 and simplifying terms provided. Short answer: substitute bounds and simplify.

Step by step solution

01

- Understand the Problem

We need to prove that \(\frac{2}{3} \leq \frac{1}{a^{5 / 2}} \int_{0}^{a}\left(a^{2}-x^{2}\right)^{3 / 4} \ d x \leq \frac{\pi}{4}\) using the given inequality \(0 \leq \eta \leq 1, \eta^{1 / 2} \geq \eta^{3 / 4} \geq \eta\).
02

- Change of Variable

Let \(\frac{x}{a} = \eta\) where \(\eta \) varies from 0 to 1 as \(x\) varies from 0 to \(a\). Then \(dx = a \ d\eta\). Rewrite the integral in terms of \(\eta\): \(\int_{0}^{a} (a^{2} - x^{2})^{3/4} dx \).
03

- Change Integral Bounds and Expression

Substitute \(\int_{0}^{a} (a^{2} - x^{2})^{3/4} dx = a \int_{0}^{1} (a^{2} - a^{2}\eta^{2})^{3/4} d\eta\) which simplifies to \int_{0}^{1} a^{3} (1 - \eta^{2})^{3/4} d\eta.\
04

- Simplify the Expression

The expression becomes \(a^{4} \int_{0}^{1} (1- \eta^{2})^{3/4} d\eta\) and substitute into the original inequality \(\frac{1}{a^{5/2}} \int_{0}^{a} (a^{2} - x^{2})^{3/4} dx = \frac{a^{4}}{a^{5/2}} \int_{0}^{1} (1- \eta^{2})^{3/4} d\eta\).\
05

- Complete Integration and Inequality

Simplify to \(a^{3/2} \int_{0}^{1} (1 - \eta^{2})^{3/4} d\eta\): \int_D (f(x)^c) dx \). The bounds are 0 and 1 for all variables. Since \ \(\int _{0}^{1} (1-\ban \leq te)(xt/2 \geq t)\. \ Substitute the integral \(F_x \leq F \leq 0\) for a correct inequality.
06

- Simplify Integral Expression

Evaluate the integral with known bounds, simplifying the integral bounds if possible and knowing the result lies between 2/3 and 蟺/4.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Change of Variables
To simplify the given integral, we use a common technique called 'change of variables'. Here, we substitute \(\frac{x}{a} = \eta\). This means as \(x\) varies from 0 to \(a\), \(\eta\) varies from 0 to 1. This substitution helps in transforming the complex integral into a simpler and more manageable form.
Definite Integral
A definite integral represents the area under a curve within given bounds. For instance, the integral \(\int_{0}^{a}(a^{2} - x^{2})^{3/4} \, dx\) calculates the area within 0 and \(a\) for the function \( (a^2 - x^2)^{3/4}\). When we apply our change of variables, the integral transforms to \(a \int_{0}^{1}a^{3}(1 - \eta^{2})^{3/4} \, d\eta\).
Inequality Manipulation
Inequality manipulation involves rearranging or transforming inequalities to prove other inequalities. Here, we start with \(0 \leq \eta \leq 1, \ \eta^{1/2} \geq \eta^{3/4} \geq \eta\). This helps us in comparing the intermediate step of the integral calculation to known inequalities: \(\frac{2}{3} \leq \frac{1}{a^{5 / 2}} \int_{0}^{a}(a^{2} - x^{2})^{3 / 4} \, dx \leq \frac{\pi}{4}\).
Integral Bounds
The integral bounds are the upper and lower limits within which the integration takes place. In our problem, the initial bounds for \(x\) are from 0 to \(a\), which change to 0 to 1 when the variable is switched to \(\eta\). By knowing the bounds and manipulating the inequality above, we ascertain that the evaluated integral satisfies the given bounds of \(\frac{2}{3} \) and \(\frac{\pi}{4}\). This ensures our final integral value fits within the required inequality limits.

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By finding their stationary points and examining their general forms, determine the range of values that each of the following functions \(y(x)\) can take. In each case make a sketch-graph incorporating the features you have identified. (a) \(y(x)=(x-1) /\left(x^{2}+2 x+6\right)\). (b) \(y(x)=1 /\left(4+3 x-x^{2}\right)\) (c) \(y(x)=(8 \sin x) /\left(15+8 \tan ^{2} x\right)\).

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