<p>i posted this on the high school life board, but i think it might fit the cafe one better:</p>
<p>i have a calculus quiz tommorrow and I am severely struggling with solving for the volume of shapes formed by intersecting graphs. I was not in class on friday when we learned it (this equals suicide for me since I am not an awesome math student).</p>
<p>Please explain how this porblem works:</p>
<p>find the volume of the solid generated by revolving the region bounded by </p>
<p>y=x, y=0, y=4, x=6</p>
<p>about the line x=6.</p>
<p>thanks in advance for taking the time to help me out here.</p>
<p>This is a shells problem (drawing the diagram will help).</p>
<p>Drawing the diagram gives you a cone shaped volume with a cylinder in the middle. I think the volume equals:</p>
<p>The formula of shells is: 2pi*[integral(a,b) (p)(h(x))]; where p is normally x (or a value minus x for empty volumes in the middle, which is in this case) and h(x) is the function value.</p>
<p>2pi<em>[integral(0,4) ((6-x)-x)]
2pi</em>[integral(0,4) (6-2x)]
2pi<em>(6x-x^2) from 0 to 4
2pi</em>((6<em>4 - 4^2)-(6</em>0-0^2))
2pi<em>(24-16)-(0)
2pi</em>(8) = 16pi</p>
<p>Hey, does your book have the answer iloveledzepplin? I'm getting something different than crypto86, but I'm too lazy to type it all out.</p>
<p>If you use the formula 2pi(r)(h)(dx) would the limits be x=[0,6] since it's a dx? (It's a dx because the strip is parallel to the axis of revolution so it cuts through the x-axis, right?) Also, if the region is rotated around x=6, how is there a hole?</p>
<p>I am not the best at math and I don't mean to criticize you crypto86. I'm just wondering for my own sake since I am also taking AP Calc. Please tell me if I am wrong.</p>
<p>This isn't a shell method because it is revolved around x=6 which would make it cone shaped. So I think you have to use a different formula. Its been a year since AB Cal though. I'm in BC, so I guess I should brush up on this stuff.</p>
<p>oh and rovee, there is no hole, i can get answers to problems like the one i listed now, but i cannot get answers to ones which do have holes. could you try and explain that to me?</p>
<p>The formula that rovee used would have been correct had this been a shell method problem. Just search for shell method and calculus on the net and you can probably find some good illustrated methods of how they work. I would suggest finding a volume program on the internet to put on your calculator for the calculator section of the ap exam.</p>
<p>Just on a side note, I think the original question could have been figured out without calculus but I will work it out to double check. I'm pretty sure you could just use the formula for the volume of a cone.</p>
<p>Oh, I guess I screwed up then. If you draw the graph from x=0 to 4 its the graph y=x, but then its a rectangle (2d) from x=4 to 6, so that's why I thought it was shells. The reference rectangle (Larson and Hostettler books use this) is paralell to the axis of revolution, so that should be shells. Interesting...</p>
<p>Right that is what I meant by solving this problem without calculus. Unless I'm misreading it, you can just translate the origin and then use R and H to find the volume of the cone.</p>