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Chapter 11: Problem 23

Below we list some improper integrals. Determine whether the integral converges and, if so, evaluate the integral. $$\int_{3}^{3} \frac{x}{\sqrt{x^{2}-9}} d x$$

Short Answer

Expert verified
The improper integral \(\int_{3}^{3} \frac{x}{\sqrt{x^{2}-9}} d x\) converges and its value is 0.

Step by step solution

01

Identify the bounds

First, look at the bound of the integral. Here, the integral is defined from 3 to 3.
02

Understand the properties of the integral

Remember one of the fundamental properties of definite integrals: if the lower bound and upper bound of a definite integral are the same, the integral equals 0. Mathematically, this is expressed as: \[ \int_{a}^{a} f(x) dx = 0 \]
03

Apply the property to the given integral

Applying this property to the integral \(\int_{3}^{3} \frac{x}{\sqrt{x^{2}-9}} d x\) gives us a value of 0, regardless of the complexity of the function being integrated.

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Key Concepts

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

Definite Integrals
Definite integrals are a central concept in calculus, representing the area under a curve from one point to another. The points are called bounds, and when the lower and upper bounds are the same, the integral results in zero. This is due to there being no interval to evaluate.
The notation for a definite integral generally looks like this:
  • \(\int_{a}^{b} f(x) \, dx\)
This reads as "the integral of \(f(x)\) from \(a\) to \(b\)." In our specific case with bounds from 3 to 3, the interval length is zero. So, according to the property of integrals, it evaluates directly to zero. This emphasizes understanding bounds is key before computing any integrals.
Convergence
Convergence is an important topic when discussing improper integrals. It tells us whether an integral has a finite value or not. Most commonly, convergence is considered in integrals with infinite limits or discontinuous integrands, but it can also relate to where an integral's bounds are different or the function behaves unpredictably.
  • If an integral converges, the area under the curve (within the limits given) is finite.
  • If it diverges, the area is infinite, and thus, the integral does not have a numerical value.
In our example, as the bounds are the same (as both 3 and 3), the question of convergence doesn't fully arise in the traditional sense, since the integral immediately simplifies to zero. No further evaluation is needed for convergence.
Fundamental Properties of Integrals
The properties of integrals form the bases that simplify solving them. One fundamental property is:
  • If the limits of integration are the same, the integral evaluates to zero: \(\int_{a}^{a} f(x) \, dx = 0\).
This comes handy in identifying that if no interval exists, there cannot be an area under the curve to measure. This property helps spot and simplify potential unwieldy calculations early.
Another important property is linearity, where sums and constant multiples can be simplified in integrals:
  • \(\int_{a}^{b} [f(x)+g(x)] \, dx = \int_{a}^{b} f(x) \, dx + \int_{a}^{b} g(x) \, dx\)
  • \(\int_{a}^{b} c\cdot f(x) \, dx = c\cdot\int_{a}^{b} f(x) \, dx\)
These properties significantly ease calculations, saving time and reducing errors.

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Most popular questions from this chapter

Set \(a_{n}=\frac{1}{n(n+1)}, n=1.2 .3, \cdots,\) and form the sequence \(S_{1}=a_{1}\) \(S_{2}=a_{1}+a_{2}\) \(S_{1}=a_{1}+a_{2}+a_{3}\) . . . \(S_{n}=a_{1}+a_{2}+a_{3}+\dots+a_{n}\) . . . Find a formula for \(S\), that does not involve adding up the terms \(a_{1}, a_{2}, a_{3}, \cdots\). HINT: Use partial fractions to write \(t$$/[k(k+1)]\) as the sum of two fractions.

Lit \(S=[\sqrt{2}, \sqrt{2+\sqrt{2}}, \sqrt{2+\sqrt{2+\sqrt{2}}}, \ldots\\} .\) Thus \(a_{1}-\sqrt{2}\) and for further subscripts \(a_{x+1}=\sqrt{2+a_{n}}\). (a) Use a graphing utility or CAS to calculate the numbers \(a_{1}, a_{2}, a_{3}, \cdots, a_{10}\). (b) Show by induction that \(a_{n}<2\) for all \(n\). (c) What is the least upper bound of \(S\) ? (d) In the definition of \(S\), replace 2 by an arbitrary positive number \(c .\) What is the least upper bound in this case?

Give the first six terms of the sequence and then give the \(n\) th term. $$a_{1}=1 ; \quad a_{n+1}=\frac{n}{n+1} a_{n}$$.

Let \(S=\left\\{a_{1}, a_{2}, a_{3}, \cdots, a_{n}, \cdots\right\\}\) with \(a_{1}=4\) and for further subscripts \(a_{n+1}=3-3 / a_{n}.\) (a) Calculale the numbers \(a_{2}, a_{3} . a_{4}, \cdots, a_{10}\). (b) Use a graphing utility or CAS to calculate \(a_{20}\) \(a_{30}, \cdots, a_{50}\). (c) Does \(S\) have a least upper bound? If so, what is it? Does S have a greatest lower bound? If so, what is it?

Use a graphing utility to draw the graph of the integrand. Then use a CAS to determine whether the integral converges or diverges. (a) \(\int_{0}^{\infty} \frac{x}{\left(16+x^{2}\right)^{2}} d x\) (b) \(\int_{0}^{\infty} \frac{x^{2}}{\left(16+x^{2}\right)^{2}} d x\) (c) \(\int_{0}^{\infty} \frac{x}{16+x^{4}} d x\) (d) \(\int_{0}^{\infty} \frac{x}{16+x^{2}} d x\)
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