How would you suggest that I try proving that an infinite region which holds a finite area, always gives a finite volume when rotated around the y-axis.
To be honest it is a question on my assignment, so I'm thinking there has to be an answer for it.
Any hints are accepted
I know that f(x)= 1/x has a infinite area from 1 to infinity but if this area is roated about the x-axis it gives a finite volume. It's actually Torricelli's trumpet.
However, I can't think of any infinite region with a finite area that when rotated about the y-axis gives an infinite volume...
true, since its under a sqrt root, you keep it under, but you put the whole expression under square root in parenthesis and take out the minus one, and then go x^2-4x. makes sense
thx guys
\int \frac {x^2}{\sqrt{4x-x^2}}dx
I just want to be sure I'm right on this, complete the square first of all so you get -\int \frac {x^2}{\sqrt{(x-2)^2-4}}dx let u=x-2 thus -\int \frac {(x+2)^2}{\sqrt{u^2-4}}dxthen let u=2sec(\theta)
hence integral becomes -8\int...
Converging lens made of ice
Would it be possible to start a fire by shaping a piece of ice ? That's the question.
I'm thinking that yes it would be possible if you shaped the ice in the form of a converging lens. Even though the lens is made of ice, the sun rays would not burn through it...
If a solid cyclinder of glass or clear plastic is placed above the words LEAD OXIDE and view from above, the LEAD appears inverted but the OXIDE does not.
The word lead is in red and the oxide is in blue. I'm thinking that since the wavelength of red light is longer, you have a left-right...
Weird, overall it gives the same answer, 4.2878 sq units, in both maple and on paper, however I don't understand how maple takes the 1/2 out of the ln.
when I integrate \int \sqrt {\frac {9x^2-36}{4}}dx I get as an answer \frac {3}{2} x{\sqrt {x^2-4}} - 6 ln({\frac {x}{2}}+\frac{\sqrt {x^2-4}}{2}) however maple gives me \frac {3}{2} x{\sqrt {x^2-4}} - 6 ln(x+\sqrt {x^2-4})
I used x=2sec(\theta) hence \frac {x}{2}=sec(\theta) so...
Find the area of the region bounded by the hyperbola 9x^2-4y^2 = 36 and the line x = 3.
I'm thinking that I have to integrate for x, so I'll have the sum of twice the area from 2 to 3.
The function will be + \sqrt {\frac {9x^2-36}{4}}
hence, the integral will be 2\int_2^3 {\sqrt {\frac...
oh damn just as i was writing this message i figured out how to apply lhopital , so simple but it escaped by grasp, its just a differentiation of an integral, thanks so much
\lim_{x \rightarrow 0} \frac{1}{x} \int_0^x {(1-tan2t)}^{\frac{1}{t}} dt
and nevermind the { in front of the 0, i couldn't figure out how to take it out, its my first time posting
I can't figure out how to start attacking this problem, do I have to intregrate by parts? if so what do I use...