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Calculus Help
I need some help on math. I have made the sketches already. I need up setting up the integrals, mainly. I can solve the math from there.
Disk and Washer Method Unit
Make a sketch and show integral.
1. Find the area bounded by the curves: [Link](Mouse over link to see full location)
a. Rotate this area around the x-axis (means find volume of solid formed)
2. Find area bounded by the curves: [Link](Mouse over link to see full location)
a. Rotate this area about x-axis (means find volume of solid formed)
b. Rotate this area about the y-axis (means find volume of solid formed)
The integrals have limits and u-substitution can be used.
Thank you!
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For the first one, it's kinda self explanatory, although it would have been easier if I could show it to you on a piece of paper. The integral when rotating around whichever axis is always multiplied by a pie (that 3.14 thingy,sorry, don't have it on the keyboard). The main formula is (pie)*S(r)^2dx. The bing "S" is the integral. Your r is always the function already given to you. So that would make it: 3.14*S(e^(x/2)+e^(-x/2))dx and you're integrating it in the interval (-1;2) because they tell you that there are two vertical lines cutting the graph into only one piece that you can rotate around the x-axis. My keyboard doesn't really allow me to put the (-1;2) on the "S" but as you know, -1 goes under the S and 2 goes above the S. From there, you can do it alone (as you already said)
Now, whoa!!!, the second one is a pretty big one, so take a deep breath:
a) If you have a calculator, graph all five of the graphs and you'll see a figure bounded by all five of them that looks like a chair, kinda, at least to me, so what you'll have to do is set up 3 integrals for part a) of the problem but first we'll find the interval at which you'll be integrating. It's different for every integral. Our first integral will be 3.14*S(-x+6)^2dx and you integrate it from x=0 to x=2 because that's where it tells you it exists. The second integral will be 3.14*S(4)^2dx and you integrate it from 2 to 4 (you're tracing the chair now if you can see that), the last integral will 3.14*S(-(x-4)^2+4)^2dx and you integrate it from 4 to 6. Now add all three results you got and you have the volume.
b) Now around the y-axis means you have to turn around the picture of the graph 90 degrees to make it more comfortable.Your first integral will be y=0 to y=4 (this time we're looking at y because we're rotating around the y-axis). You can of course see the graph of y=-(x-4)^2+4 but you have to turn it in terms of x, meaning that in terms of x, the equation would be x=(4-y)^(1/2)+4. Now you first integral is 3.14*S((4-y)^(1/2)+4)^2dy and you integrate from 0 to 4. Your second integral will be integrated in the interval from 4 to 6. The equation (y=-x+6) again has to be in terms of x, meaning x=6-y.
So you second integral would be 3.14*S(6-y)^2dy and you integrate it from 4 to 6. Add the two results together and that should be your answer. I really do hope you managed to get something from all of this, cause without a piece of paper and me actually showing it to you there, I think that in your place, I would never have gotten it through the net. Hope I helped, if you have any questions, ask me :)
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