Thread: Antiderivative Help

I am getting the concept of anti derivatives but when constants are involved I am having trouble for some reason. I have the following homework question

If a diver of mass m stands at the end of a diving board with length L and linear density p then the board takes on the shape of a curve y = f(x) where

EIy′′= mg(L -x) + 1/2pg(L - x)²

E and I are positive constants that depend on the material of the board and g(< 0_ is the acceleration due to gravity.

I need to find an expression for the shape of the curve. To do this I would need to find the anti derivative of EIy′′ which would be EIy′ and then find the anti derivative of EIy′ to get ELy which is my f(x) correct? If so how do I start to anti differentiate EIy′′? Any help would be appriciated. Thanks

Originally Posted by fishguts

I am getting the concept of anti derivatives but when constants are involved I am having trouble for some reason. I have the following homework question

If a diver of mass m stands at the end of a diving board with length L and linear density p then the board takes on the shape of a curve y = f(x) where

EIy′′= mg(L -x) + 1/2pg(L - x)²

E and I are positive constants that depend on the material of the board and g(< 0_ is the acceleration due to gravity.

I need to find an expression for the shape of the curve. To do this I would need to find the anti derivative of EIy′′ which would be EIy′ and then find the anti derivative of EIy′ to get ELy which is my f(x) correct? If so how do I start to anti differentiate EIy′′? Any help would be appriciated. Thanks

The first thing you should do is write out what y'' equals, with everything expanded:
y''=(1/EI)*mgL - (1/EI)*x + (1/2)*(1/EI)*pg*L^2 - (1/2)*(1/EI)*pg*(2L)*x + (1/2)*(1/EI)*pg*x^2.

This looks like a big mess, but everything except for x is a constant. So let me rewrite this as
y''=A+Bx+C+Dx+Fx^2

The antiderivative of the sum is equal to the sum of the antiderivatives, so just take antidervivative of each term individually.

y'=Ax+(1/2)Bx^2+Cx+(1/2)Dx^2+(1/3)Fx^3 + K (a new constant)

Take the antiderivative of each of these terms, and you are done! (just plug in the more complicated looking constant values)

Originally Posted by robeuler

The first thing you should do is write out what y'' equals, with everything expanded:
y''=(1/EI)*mgL - (1/EI)*x + (1/2)*(1/EI)*pg*L^2 - (1/2)*(1/EI)*pg*(2L)*x + (1/2)*(1/EI)*pg*x^2.

shouldn't the second term be -mgx*(1/EL) instead of - (1/EI)*x ?

Originally Posted by fishguts

shouldn't the second term be -mgx*(1/EL) instead of - (1/EI)*x ?

yeah I probably made a mistake with the algebra. did you get the calculus part though?

Originally Posted by robeuler

yeah I probably made a mistake with the algebra. did you get the calculus part though?

Yes I got the calculus part and is its a really long expression but thanks for the help!