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Chapter 6: Problem 11

For Activities 5 through \(16,\) evaluate the improper integral. $$ \int_{2}^{\infty} \frac{1}{\sqrt{x}} d x $$

Short Answer

Expert verified
The integral diverges to infinity.

Step by step solution

01

Understand the Improper Integral

This integral is improper because it extends to infinity, so the upper limit is infinite. We will need to use limits to evaluate it.
02

Set Up the Limit

Convert the integral from an improper integral to a limit: \[ \lim_{b \to \infty} \int_{2}^{b} \frac{1}{\sqrt{x}} \, dx \]
03

Find the Antiderivative

The integrand is \( \frac{1}{\sqrt{x}} = x^{-1/2} \). Find the antiderivative: \[ \int x^{-1/2} \, dx = 2x^{1/2} + C \] where \( C \) is the constant of integration.
04

Evaluate the Definite Integral

Evaluate the definite integral from 2 to \( b \) using the antiderivative: \[ \int_{2}^{b} \frac{1}{\sqrt{x}} \, dx = 2x^{1/2} \bigg|_{2}^{b} = 2b^{1/2} - 2(2)^{1/2} \] Simplifying, we get: \[ 2b^{1/2} - 4 \]
05

Take the Limit as b Approaches Infinity

Now, evaluate the limit: \[ \lim_{b \to \infty} (2b^{1/2} - 4) \] Since \( b^{1/2} \to \infty \), \( 2b^{1/2} \to \infty \), so:\[ \lim_{b \to \infty} (2b^{1/2} - 4) = \infty \]
06

Conclusion: Divergence

Since the result is \( \infty \), this improper integral diverges. The integral does not converge to a finite value.

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Key Concepts

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

Infinite Limits
When dealing with improper integrals, one of the challenges is handling infinite limits. These integrals involve limits that extend infinitely. In the exercise provided, the integration limits stretch from a finite value (2) up to infinity. This creates a condition where traditional calculus techniques need adjustment.

To evaluate such integrals, we use a limit process.
  • We reformulate the integral as a limit problem.
  • Instead of integrating up to infinity directly, we integrate up to a finite boundary 'b' and then take the limit as 'b' approaches infinity.
This approach helps in systematically handling the complication of the upper limit being infinite. It breaks down the problem into manageable parts, enabling us to utilize the familiar rules of definite integration.
Antiderivatives
In calculus, finding an antiderivative is a crucial step in solving integrals. For the integral in the exercise, we're asked to find the antiderivative of the function \( \frac{1}{\sqrt{x}} \). Converting this function into some form suitable for integration is key.

The given function \( \frac{1}{\sqrt{x}} \) can be rewritten in the exponent form as \( x^{-1/2} \). To find its antiderivative, we apply basic antiderivative rules:
  • We add 1 to the exponent, changing \( x^{-1/2} \) to \( x^{1/2} \).
  • We then divide by the new exponent, \( \frac{1}{1/2} = 2 \).
Thus, the antiderivative of \( \frac{1}{\sqrt{x}} \) is \( 2x^{1/2} + C \), where \( C \) is the constant of integration. Understanding this step is essential in transitioning from the integrand to a usable form for evaluating limits.
Divergence of Integrals
The concept of divergence is critical when analyzing improper integrals. In our exercise, after calculating and evaluating the integral with upper limits as infinity, we encounter a result that trends towards infinity itself.

The divergence or convergence of an integral refers to whether the integral results in a finite value or grows indefinitely.
  • A convergent integral safely "settles" to a finite number.
  • A divergent integral, like this one, does not settle and thus tends to infinity or remains undefined.
The given integral \( \int_{2}^{\infty} \frac{1}{\sqrt{x}} \, dx \), after evaluation, leads to a limit of \( \infty \), suggesting that the integral diverges. This indicates that there isn't a finite area under the curve from 2 to infinity, shaping our understanding that some integrals, while defined in theory, span an endless value.

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Most popular questions from this chapter

Frozen Yogurt Sales Let \(x\) represent the amount of frozen yogurt (in hundreds of gallons) sold by the G\&T restaurant on any day during the summer. Storage limitations dictate that the maximum amount of frozen yogurt that can be kept at G\&T on any given day is 250 gallons. Records of past sales indicate that the probability density function for \(x\) is approximated by \(y(x)=0.32 x\) for \(0 \leq x \leq 2.5\) a. What is the probability that on some summer day, G\&T will sell less than 100 gallons of frozen yogurt? b. What is the mean number of gallons of frozen yogurt G\&T expects to sell on a summer day? c. Sketch a graph of \(y\) and locate the mean on the graph and shade the region whose area is the answer to part \(a\).

Tree Height The height, \(h,\) in feet of a certain tree increases at a rate that is inversely proportional to time, \(t\) in years. The height of the tree is 4 feet at the end of 2 years and reaches 30 feet at the end of 7 years. a. Write a differential equation describing the rate of change of the height of the tree. b. Give a particular solution for this differential equation. c. How tall will the tree be in 15 years? What will happen to the height of this tree over time?

Write an equation or differential equation for the given information. In mountainous country, snow accumulates at a rate proportional to time \(t\) and is packed down at a rate proportional to the depth \(S\) of the snowpack. Write a differential equation describing the rate of change in the depth of the snowpack with respect to time.

Plow Patents The number of patents issued for plow sulkies between 1865 and 1925 was increasing with respect to time at a rate jointly proportional to the number of patents already obtained and to the difference between the number of patents already obtained and the carrying capacity of the system. The carrying capacity was approximately 2700 patents, and the constant of proportionality was about \(7.52 \cdot 10^{-5} .\) By 1883,980 patents had been obtained. (Source: Hamblin, Jacobsen, and Miller, \(A\) Mathematical Theory of Social Change, New York: Wiley, 1973 ) a. Write a differential equation describing the rate of change in the number of patents with respect to the number of years since 1865 b. Write a general solution for the differential equation. c. Write the particular solution for the differential equation. d. Estimate the number of patents obtained by 1900 .

Let \(w\) with input \(x\) be a uniform density function with \(a=4\) and \(b=20\) a. Write \(W\), the corresponding cumulative distribution function. b. Use both \(w\) and \(W\) to calculate the probability that \(x \leq 5.8\) c. Use \(W\) to calculate the probability that \(15
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