a) Calculate the are between the lines and and the parabola
b) Calculate the rotation volume of the body that comes when the curve rotates around the x-axis
Is there a way to post a coordinate system here with curves and stuff in it?
Ok so i find the intersection points by putting and
When I get the x-coordinates I can easily find the y-coordinates. So I have the intersections (1,1),(2,0) and (-2,-2). So basically the x-values will be the intervals I´ll be using?
Somebody told me to divide the figure in two by drawing a vertical line crossing (1,1) in order to more easily calculate the two areas.Is this right?
I´ll start with the first area:
And then i put in the interval values, first 2 and subtract the result with the result when putting in 1?
which is ??
The second area is calculated in similar fashion?
Which would be
Added together ?
In this case we want .
The formula for integration by parts is . Here's my suggestion: for choose the most complicated portion of the integrand that you know how to integrate. In our case, is the most complicated factor, but it is integrated easily. So we would have
Once you make your substitutions, you will end up with
This integral, , will require another application of integration by parts.
In general, here's some tips for choosing your and :
For integrals of the form (and other trig functions), let and .
Similarly, for integrals of the form , let and .
For the integrals or , let or , and let . Use integration by parts twice, and you will actually end up with an equation that can be solved for the desired integral.
For integrals like or , let be the entire integrand with .
You can also do some searching around on Google. I found, for example: MathGV and gnuplot.
If you have Windows XP, you can also look at the Windows XP PowerToys. They have a very nice calculator that can do graphing.
And for simple graphs, you can often create them by hand in an image editor.
Anyway here´s my attempt (I´ll leave the and for later and only try the integration part:
And then putting it in the formula we get:
You should get
Now, I do suspect you should have been given some kind of interval. Although you can have a surface of revolution created from an infinitely long curve and still have it enclose a finite volume (see e.g. Gabriel's Horn for a surprising example), this volume would definitely be infinite if you evaluated it over the whole real line.
You think you can help me with another thing. Since I write all my homework on a computer it would be nice to write it with some fancy software. I´ve been using word2007 which is okay but there are some features I miss. Like the big brackets for instance. This Latex thing which is totally new to me is superb. Im in love with it. I mean I can do basically anything with it, well not yet but after I´ve used it for a while
Word 2007 would be excellent if I could "insert latex box" where ever in the document.
Download a port: MiKTeX is a good one; you can find others at the CTAN (they also have a nice introduction and help for beginners).
If submitting your papers online, you can use pdfTeX (included with most distributions) to get a PDF file.
It may take you a little while to get up and running, and your writing might be a little slow until you get the hang of it, but it is worth it if you want professional, quality output.