Skip to main content

Find the surface area of revolution of `y = 1/x` from `xgt=1` The setup of the integral is SA = `2pi int_0^oo(1/x*sqrt(1+1/x^2))dx` Then, you need...

There seems to be a discrepancy in your question: if `x >=1` , then the limits of the integral should be 1 and infinity. This means it would be an improper integral and it would not converge (as you will see after we take the integral). I am going to take the indefinite integral - without the limits - and the appropriate limits can be plugged in later.

Before taking the integral, let's simplify the radical expression under the radical:


`sqrt(1 +1/x^2) = sqrt((x^2+1)/x^2) = sqrt(1+x^2)/x` .


The integral then becomes (omitting the coefficient of `2pi` for now):


`int sqrt(1+x^2)/x^2 dx` . Notice that it contains the radical expression `sqrt(1 + x^2)` . The traditional way to take such integrals is by using trigonometric substitution


`x = tant` . Then, `dx = sec^2tdt` and `t = arctanx` .


The advantage of this substitution is that it allows to simplify the radical expression using the trigonometric identity `tan^2t + 1 = sec^2t` .


`sqrt(1+x^2) = sqrt(1+tan^2t) = sqrt(sec^2t) = sect` .


Plugging the above into the integral, we get


`int (sect)/(tan^2t) sec^2tdt` . This can be rewritten again using the same trigonometric identity as


`int (sect)/(tan^2t) (tan^2t + 1)dt = int sectdt + int (sect)/(tan^2t) dt` .


The first integral equals


`int sect dt = ln|sect + tant|` . Substituting the original variable x back, we get `ln|sqrt(1+x^2) +x|` .


To take the second integral, use `sect = 1/(cost)` and `tan^2t = (sin^2t)/(cos^2t)` :


`int (1/(cost))/((sin^2t)/(cos^2t)) dt = int (cost)/(sin^2t) dt` . Here, we can use substitution


u = sint. Then, du = costdt and the integral becomes


`int (du)/u^2 = -1/u = -1/(sint)` . In terms of x = tant,


`sint = tantcost = (tant)/(sect) = x/sqrt(1+x^2)` .


Putting it all together, we see that the original integral (without the limits) equals to


`2piln|sqrt(1+x^2) + x| - 2pisqrt(1+x^2)/x` + C (C is an arbitrary constant.)

Popular posts from this blog

In chapter one of The Great Gatsby, what advice does Nick's father give him? How does this make him a good person to tell this story?

Nick says that his father advised him that, before "criticizing anyone," he "remember that all the people in this world haven't had the same advantages" as Nick.  As a result, Nick claims that he is "inclined to reserve all judgments," presenting himself to the reader as a fair and dispassionate arbiter of character, and thus, a reliable narrator.   The problem is that Nick immediately reveals himself as anything but reliable, as he then launches... Nick says that his father advised him that, before "criticizing anyone," he "remember that all the people in this world haven't had the same advantages" as Nick.  As a result, Nick claims that he is "inclined to reserve all judgments," presenting himself to the reader as a fair and dispassionate arbiter of character, and thus, a reliable narrator.   The problem is that Nick immediately reveals himself as anything but reliable, as he then launches into a discussion of how pe...
Post a Comment
Read more

How did the United States become an imperial power?

"Imperial power" is a bit of a vague label. It tends to mean several things at once, so let's unpack it. In the sense of "this country was built on conquest by force," the "imperial power" part of America actually predates the United States proper. The territories that would become the United States were imperial colonies, established by the great European empires of the 17th and 18th centuries. Much of the US Constitution and American governance generally goes back to England, history's largest and most successful imperial power, but vital aspects of American culture come from other imperial powers, such as France and Spain. Much of American culture comes from sources other than the old empires, but they were key influences on what the United States became. In the sense of "this country treats conquest by force as a fundamental component of its culture, economy and politics," the United States has always been an imperial power. Even earl...
Post a Comment
Read more

How and why does James Gatz become Jay Gatsby? Describe the young Gatsby/Gatz.

James Gatz, a poor Midwestern boy of probable Jewish lineage, becomes Jay Gatsby, a presumed WASP and wealthy socialite, when he moves to New York City and acquires his fortune. It is wealth that has allowed Gatz to transform himself into Gatsby. However, those who know his background (e.g., Daisy and Tom Buchanan) never allow him to forget that he is nouveau riche -- that is, an upstart who has just recently made his fortune,... James Gatz, a poor Midwestern boy of probable Jewish lineage, becomes Jay Gatsby, a presumed WASP and wealthy socialite, when he moves to New York City and acquires his fortune. It is wealth that has allowed Gatz to transform himself into Gatsby. However, those who know his background (e.g., Daisy and Tom Buchanan) never allow him to forget that he is nouveau riche -- that is, an upstart who has just recently made his fortune, whereas they arose from well-to-do families. Gatz became Gatsby through determination and discipline. At the end of the novel, the narr...
Post a Comment
Read more