/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 Determine whether the improper i... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 37

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{2} \frac{1}{\sqrt[3]{x-1}} d x $$

Short Answer

Expert verified
The integral converges, and its value is 0.

Step by step solution

01

Establish the two bounds

Since there's a discontinuity at \(x=1\), the problem breaks down into two integrals: \( \int_{0}^{1} \frac{1}{\sqrt[3]{1-x}} d x \) and \( \int_{1}^{2} \frac{1}{\sqrt[3]{x-1}} d x \) We should evaluate both of these.
02

Evaluate the first integral

Let's evaluate the first integral \( \int_{0}^{1} \frac{1}{\sqrt[3]{1-x}} d x \). This is an indefinite integral, but as x approaches 1 from the left, the denominator approaches 0, making the integral improper. A possible solution for this is to use the limit definition of an improper integral: \( \lim_{a \to 1^-} \int_{0}^{a} \frac{1}{\sqrt[3]{1-x}} d x \). The antiderivative is \(-3(1-x)^{2/3}\) by using the power rule, substituting and simplifying, this limit comes out to be \(-3\).
03

Evaluate the second integral

Now, let's evaluate the second integral \( \int_{1}^{2} \frac{1}{\sqrt[3]{x-1}} d x \). Similar to the steps we took to solve the first integral, we'll work with the limit definition of this improper integral: \( \lim_{b \to 1^+} \int_{b}^{2} \frac{1}{\sqrt[3]{x-1}} d x \). Also, the antiderive of \( \frac{1}{\sqrt[3]{x-1}} \) is \(3(x-1)^{2/3}\), substitituting and simplifying, the limit comes out to be 3.
04

Add the results

The final step is to add both results together: \(-3+3=0\)

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Most popular questions from this chapter

True or False? In Exercises 67-70, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(p(x)\) is a polynomial, then \(\lim _{x \rightarrow \infty}\left[p(x) / e^{x}\right]=0\).

Sketch the graph of \(g(x)=\left\\{\begin{array}{ll}e^{-1 / x^{2}}, & x \neq 0 \\ 0, & x=0\end{array}\right.\) and determine \(g^{\prime}(0)\).

Laplace Transforms Let \(f(t)\) be a function defined for all positive values of \(t\). The Laplace Transform of \(f(t)\) is defined by \(F(s)=\int_{0}^{\infty} e^{-s t} f(t) d t\) if the improper integral exists. Laplace Transforms are used to solve differential equations. Find the Laplace Transform of the function. $$ f(t)=e^{a t} $$

Let \(f^{\prime \prime}(x)\) be continuous. Show that $$ \lim _{h \rightarrow 0} \frac{f(x+h)-2 f(x)+f(x-h)}{h^{2}}=f^{\prime \prime}(x) $$

Find the arc length of the graph of \(y=\sqrt{16-x^{2}}\) over the interval [0,4]
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