An integral in which one of the limits of integration is infinity is an improper integral. Because we cannot find the definite integral using infinity (since it is not an actual value), we will need to rewrite the improper integral as a limit, shown below:
int_1^oo 6/(x^4)dx=lim_(n->oo)int_1^n 6/(x^4)dx
We can now take the integral first and the limit second. We can find the integral using the power rule:
lim_(n->oo)int_1^n 6/(x^4)dx=lim_(n->oo)int_1^n 6x^-4dx=lim_(n->oo)[6*x^-3/-3]_1^n=lim_(n->oo)[-2/x^3]_1^n
Now we simply need to evaluate the limit. As n approaches infinity, the fraction will approach 0:
lim_(n->oo)-2/n^3-(-2)/1^3=lim_(n->oo)-2/n^3+2=0+2=2
Thus the integral converges to a value of 2.
Sunday, July 21, 2019
Calculus of a Single Variable, Chapter 8, 8.8, Section 8.8, Problem 18
Subscribe to:
Post Comments (Atom)
Why is the fact that the Americans are helping the Russians important?
In the late author Tom Clancy’s first novel, The Hunt for Red October, the assistance rendered to the Russians by the United States is impor...
-
There are a plethora of rules that Jonas and the other citizens must follow. Again, page numbers will vary given the edition of the book tha...
-
Lionel Wallace is the subject of most of "The Door in the Wall" by H.G. Wells. The narrator, Redmond, tells about Wallace's li...
-
In the late author Tom Clancy’s first novel, The Hunt for Red October, the assistance rendered to the Russians by the United States is impor...
-
The poem contrasts the nighttime, imaginative world of a child with his daytime, prosaic world. In the first stanza, the child, on going to ...
-
Robinson Crusoe, written by Daniel Defoe, is a novel. A novel is a genre defined as a long imaginative work of literature written in prose. ...
-
In Celie's tenth letter to God, she describes seeing her daughter in a store with a woman. She had not seen her daughter since the night...
-
Let's start with terms: "expected value" means the average amount that you would win or lose over a large number of plays. The...
No comments:
Post a Comment