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The rate of change of the vertical deflection $y$ with respect to the horizontal distance $x$ from one end of a beam is a function of $x$ For a particular beam, the function is $k\left(x^{5}+1350 x^{3}-7000 x^{2}\right)$ where $k$ is a constant. Find $y$ as a function of $x$ if $y=0$ when $x=0$

Calculus 2 / BC

Chapter 26

Applications of Integration

Section 1

Applications of the Indefinite Integral

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Okay, so we're told that dy dx is equal to k, multiplied by X. To the fifth plus 13 50 times X. To the third minus 7000 times X squared. And we want to find a function for why given that Y is equal to zero when X is equal to zero. So what we can do is we can just integrate both sides. Here can say the integral of the Y is equal to the integral of K. X. to the 5th plus 13 50 time six, third minus 7000 times X squared dx. So now we can just find this integral. So we can take our constant K out. So you'd have this is equal to K multiplied by the integral fix. The fifth plus 13 50 Times X to the 3rd minus 7000 times X squared. And this is all multiplied by dx. And so now we can do is we can find the integral of each of these terms. Um And so we would have, this is equal to k. Multiplied by X to the 6th, divided by six. We just add one to the exponents and divide by that exponents. We're gonna do the same thing here. We can treat 1350 as a constant, so we won't change that, we still have 1350 And this is multiplied now by X to the 4th and divided by four. And we have again we can do the same thing here. So we have we keep our constant 7000. We raise our exponent one and we divide by the resulting exponent and then we also have to remember plus C. So now we have why is equal to K Multiplied by X. The 6th, guided by six plus 13 50 Times X. To the 4th, divided by four minus 7000 times X to the third, divided by three plus C. So this is equal to y. So now we can use the fact that when X is equal to zero, Y is equal to zero. To figure out what C. Is equal to. So we have zero, it's equal to k. Multiplied by 0-6, divided by six is just zero, And then 0 to the 4th Multiplied by 1315, divided by force also zero. And this term is also going to go to zero and X equal to zero suite of K multiplied by zero plus zero minus zero plus C. So then zero is equal to K. Multiplied by C. So then C. is equal to zero. So now we have our equation for why which is equal to K. Multiplied by X to the 6th, divided by six plus 13 50 Multiplied by X. to the 4th, divided by four In -7000 times X. To the third, divided by three.

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