Find the volume of the solid generated by revolving the shaded region about the x-axis. 5x+2y =10

Question

jigglypuff314 · Answer

Hello and Welcome to OpenStudy! :)
do you have a picture of where the shaded region would be?

anonymous · Answer

Hi, and thanks!

anonymous · Answer

attachment

jigglypuff314 · Answer

I'm guessing you're in calculus?

anonymous · Answer

Yes, Math 104

jigglypuff314 · Answer

alright
so if you rotate the shaded region around the x axis
what do you think you would get?
a washer? or cylinder thing?

anonymous · Answer

Couldn't you just calculate the area of the triangle, and multiply that by$$2\pi r$$to get the volume? I hate shaded regions

anonymous · Answer

Would that work Embryo? I hadn't thought of it... Hmm

anonymous · Answer

And to jigglypuff314, I have this so far:

anonymous · Answer

$$V \int\limits_{0}^{2} \pi (R(x))^2 dx$$  $$R(x) = (10-5x)/2$$    $$5x + 2(0) = 10$$

jigglypuff314 · Answer

@Embryo I'm not getting the same answer as I would have if I did it the calculus way...

anonymous · Answer

Then dangit, i really hate shaded regions

jigglypuff314 · Answer

jghost, the first two steps look right to me
I'm not sure what the third thing you have is :/
just plug in  what you have for R(x)
into the integral and solve

anonymous · Answer

Sorry, the third thing was how I got the Upper Limit of Integration to be 2... but I'm having trouble solving :(

anonymous · Answer

@Embryo, revolving the region would give you a cone. You could use the formula for a cone's volume to avoid the integration method.

parthkohli · Answer

Here's a non-calc way.

By a little imagination, the volume of this thing is exactly the half of the volume of a cylinder with radius 5 and height 2.$$\pi \times 25 \times 2 \times \dfrac{1}{2}$$$$= 25\pi$$

parthkohli · Answer

Is that the right answer...?

anonymous · Answer

$$V=\pi r^2\frac{ h }{ 3 }$$where r is the radius, being 5, and h= the height from the base to the tip, which is 2, like that @SithsAndGiggles ?

jigglypuff314 · Answer

@Embryo yes :) that is the answer I got from the integration way

anonymous · Answer

@Embryo, yeah

anonymous · Answer

So @JGhost you now have multiple ways of solving this, pick your favorite and roll with it

jigglypuff314 · Answer

$$\large \pi \int\limits_{0}^{2} (-2.5x + 5)^2~~ dx$$ would be the integration way, which you can/should just plug into calculator :P

anonymous · Answer

Hmm I went with the calc method and got 50pi/3 as the ans?

parthkohli · Answer

OK, then go with the calc method.

jigglypuff314 · Answer

correct! :)

parthkohli · Answer

This makes way for another kind of curiosity.

Why was the non-calc way wrong?

parthkohli · Answer

Ah, I see... never mind.

anonymous · Answer

Sweet! The system accepted it as correct! Thanks peeps. @jigglypuff314 @Embryo @ParthKohli @SithsAndGiggles

jigglypuff314 · Answer

glad we could help ^_^