Integration

[hine0200]

Please help me with a couple of problems!!

1. The centers of two disks with radius 1 are 1 unit apart. Find area of union of two discs.

2. Is the area between curves y= x + (1/x^2) and y= x - (1/x^2) finite or infinite? If finite, what is the area? And if we were to rotate around the x-axis what is the volume, if the area is finite?

3. Show that the limit as t approaches infinity of the integral of sinx from -t to t equals zero. Also show that the integral of sinx from neg. infinity to pos. infinity is divergent.

Any and all help getting me started with these is appreciated. Thanks

 

[tkhunny]

Any and all indication that you actually have any idea at all, that you are attending class, that you have made some sort of effort to succeed would be great.

Answer the first part of #3. It is a trivial answer going to the nature of the sine function.

For #2, since 1/x^2 is always positive, the distance between the two functions is (x+(1/x^2))-(x-(1/x^2)) = 2/x^2. What say you of this integral. It has problems around x = 0. Can you overcome them?

For #1,
1) Find some equations. I recommend x^2 + y^2 = 1 and (x-1)^2 + y^2 = 1
2) Exploit symmetries. Take y > 0 and multiply by 2.
3) Exploit symmetries. Take x > 1/2 and multiply by 2 again.
4) The steps above shoudl lead you to the shared area.
5) Calculate the area of the two disks.
6) The rest is arithmetic.

 

[hine0200]

Sorry, number two is to find the area for x greater than or equal to one.

For number three, I worked the first part out to the point where I get the limit of cos(-t) - cost as t approaches infinity and i do not know how to show that that equals zero

 

[tkhunny]

#3 - What? There is nothing to "work out". What do you know about the sine function that will answer this question? Try evaluating this and see if it gives you a clue...

\(\displaystyle \int_{-1}^{1}x\;dx\)

#2 - Don't just give more information. Also show your additional efforts. Note: Something fishy about the wording. Why do we care about the volume ONLY IF the AREA is finite?

\(\displaystyle \int_{1}^{\infty}\frac{2}{x^{2}}\;dx\)

Well, does it converge?

 

[hine0200]

So for number three, the first part is zero due to the symmetry of the sine function (and no calculations are necessary). Then for the second part of number two, it is divergent because after you integrate you get the limit of cos(-t) - cost as t approaches infinity (where the limit does not exist)?

 

[tkhunny]

You didn't quite find the right word, although "symmetry" is good. It's just a little too generic. f(x) = sin(x) is an ODD function, so that f(x) = -f(-x). This is why the integral is zero over any finite interval centered on x = 0. You will find it to be the case for ANY "odd" function.

The infinite limit really doesn't have much to do with the cosine, although this may convince your thought process as far as a "conjecture". In order for an integral to exist, the function must approach a limit. What limit does f(x) = sin(x) approach as x increases without bound. It is bounded, to be sure, but does it approach a limit?

Just for a better point of rerference, what class are you taking for these problems? Obviously some sort of "calculus", but is there a particular emphasis? What major requires it? How do you think it is going for you?