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April 23rd, 2014, 06:12 AM  #1 
Senior Member Joined: Dec 2013 Posts: 137 Thanks: 4  Volume Problem (showing lots of work)
Here is the problem... Gyazo  67aaed27d46cd857a41289d1def5d9d7.png Here is what I did. To find the area of the crosssectional area A(y), I solved the 2 given equations y = a root x and y = a x ^b for y and got x = (y/7)^2 and x = (y^(1/5))/7^(1/5) now, using the washer method deal I go A(y)= (piR^2  pir^2)dy and came up with R = (1  y^2 / 49) and r = (1  y^(1/5)/7^(1/5) now putting that all together and Integrating from 0 to 7 I quickly see that when it's time to plug and chug the 0 is going to cancel all equations thus I will have this  0 = that so I essentially just need to lay these eq's out as pi INTEGRAL (R^3  r^2)  (0) dy = now is the point right before integrating where we combine equations and I am getting... (2y^(2)/49) + (y^4)/2401 + 2y^(1/5)/(7^1/5)  y^(2/5)/7^(2/5) from 0 to 7 (again, we are just going to be plugging in the b =7 value when integrating because plugging in zero will just give 0 , thus it is unnecessary. Now, this is the point when I started to get excited because I started to see that plugging the b value of 7 into y and CHUGGING! as you guys say ... I quickly saw that everything was canceling so nice ...almost like I had stacked a million dominos and they were all coming down in 30 secs (haha u math folks like that analagy..lol) Anyways, very unfortuneately, I ended up with Pi (0) thus giving me 0 and I am sure that volume of this problem is not zeor. So help you guys! 
April 23rd, 2014, 08:55 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,271 Thanks: 2434 Math Focus: Mainly analysis and algebra 
I think that you forgot to integrate!

April 23rd, 2014, 06:55 PM  #3 
Senior Member Joined: Dec 2013 Posts: 137 Thanks: 4 
V8 I think you are def right. So, I did Integrate now and got 17pi/5 Does that look right V8? 
April 23rd, 2014, 07:11 PM  #4 
Senior Member Joined: Dec 2013 Posts: 137 Thanks: 4 
V8 it is right so don't trouble yourself with it. thanks very much for pointing out the fact that I didn't integrate. I went back and did that and got the right answer so thanks for the help 
April 23rd, 2014, 08:36 PM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,271 Thanks: 2434 Math Focus: Mainly analysis and algebra 
No problem. I'm glad to be of service.

April 24th, 2014, 10:48 AM  #6 
Senior Member Joined: Dec 2013 Posts: 137 Thanks: 4 
V8 if you are still around... here is an almost identical problem but this time a different region with slightly different functions. I'm confused again on how to set it up (i.e., whether to use y or x or if I'm simply overthinking the problem) Gyazo  bed125b39432da10902bbffa8e8c86b4.png 

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