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A water storage tank has the shape of a cylinder with diameter 10 ft. It is mounted so that the circular cross-sections are vertical. If the depth of the water is 7 ft, what percentage of the total capacity is being used?
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Calculus 2 / BC
Techniques of Integration
March 2, 2022
Why did you go from 25l( theta + Sin(theta)Cos(theta)) to 25l (theta -Sin(theta)Cos(theta))?
University of Nottingham
Idaho State University
In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative.
The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis.
The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.
In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.
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Ah, water storage tank has the shape of a cylinder with diameter. Ten. So that means that the radius is fire. Write that on the side It's mounted so that the circular cross sections are vertical. So in other words, it's the tank is lying on inside. And then the cross sections looked just like this lid over here, the circle on the edge of the screen, there for nickel and they're circular. So this is how we know that it's mountains the correct way. And then the depth of the water was seven. We like to know what's the percentage of the total capacity that's being used? So this expression that we want this is just the volume of the water in the tank divided by the volume of the tank, which is just a we'LL see. It's just that I think is the cylinder. So the volume of the tank is just the volume of this black film there on the left. So first we'll find the volume of the water so many ways to do this. Probably the easiest way is to notice that this volume of this water right here since all of these cross sections of the same if he could do to the fact that the water is on the level. Right? So any time you cross through this thing, all the cross sections looked like this live. It's a circular circle. And then we have this blue region that goes all the way up here in the water into the water line. So we could do is just find the area of this lid and then multiply the whole thing by the length of the cylinder. We'LL just call that no plans for the height, but everyone called. So volume of water volume some water. We'Ll just be this area out here of this disc. So the area of the water times l and we can go ahead and express this as an integral So the way to do this here One way to do this is to just look at this circle this lit out here at the edge. Let's go ahead and put some X and y axes on there. So right there the center of the lid. That's the origin. We're told that the dented the water seven. No, If this is zero at the origin, the radius is five So that puts me in a negative five here and at the height of seven. That means we go from negative five all the way up until two. So if we want to integrate with respect, sir, why those air are y bounds also for integrating with respect. So why, that means that we're integrating Berkeley. So our cross sections here Wilby these lines somewhere generating top mind his bottom or in this case, right minus left. And then from negative five all the way up to two. So we should do. Here's find the equation of the circle has radius five and centered at the ocean. So there's the equation and solve that for X. And this gives us the two equations for the left and right half of the circle. So the right half over here, I'm tracing over and read this right half the equation for that And for the left side, that will be the negative radical. So the side over here on green, that equation is X equals negative. Route twenty five minus y squares. So now we could do right minus left. So this over here now becomes integral. Negative five to two and in the left, the wrecker minus, then left curve. So we can go ahead and go to the next page and simplify this. We have to enroll. Negative five to two, twenty five minutes. Why square do? And now we have a new integral to work on. And also, this was just a so it's good and actually multiply this by that that l the link that's on the right hand side. So we just keep multiplying by hell. But really, our work here is under is integral. Looking at the inside of the radical, we should probably go ahead and take Y b five Signed Ada then d y five course idea. Yeah, this is a definite integral. So you should always try to switch the limits in terms of the Newbury Bowl. Dana here. But if you do that, you won't be able to find values of data that you're familiar with. Indiana Circle. So what we'LL do here is well, just denote the new limits by Andy. Hey, so first also looks good in simplify this, the radical becomes twenty five minus twenty five science where pull out that twenty five and take the square root you get a five one minus sign square, use Taggart identity and then take the square root. So we'LL have to integral a to B five course idea then the y, which was also fat ko Santa. And then let's just put that out of here somewhere with her, too. So two times milk Sorry, Now God and multiply this out. So we have fifty times two fifty l integral A b Now here we have co science Where these two co signs being multiplied. Let's use the fact that cool science where one plus co sign tooth data all over too. That's the half angle identity. So pull that too. Out outside the Integral one plus co sign to theatre Here's twenty five l And now we could integrate this data scientific data over to a TV And now here for the science. You there now we'LL want to simplify this. Yes, so right it is to side data co signed data That's the double angle formula for sign and cross off those twos. So we have twenty five ill data, plus ScienceTimes goes on, maybe score the next age running out of room here. So previously we had this expression here and now will want to go ahead and use the triangle to write everything back in terms of X. It's a co sign there, that ready triangle. So here are true, son. It's equivalent to saying why over five equal sign data. So let's go ahead and take in our triangle sign is why over hype on news of this for a five there. And then you could find that this missing side age using today during zero equals High Palm Square and insult for each. Now we could find Sinan co sign. So twenty five l data. You could find it from this equation. Here, take sign in verse. Both sides. You could sign inverse whatever. Five data. So that goes here minus sign. Which is why, over five times co sign, which is h over five. It's a H up top and then fight again on the bottom. Now, since we're back in our variable, why we can go ahead and use those original limits. Why was going from negative five to two? So this is why we didn't have to find Andy because we were willing to get back into the variable life. It's good and plug in these numbers to first. Yeah, so it's for twenty five in that bottom and then radical twenty one. And then when we played your negative five sign in verse Negative one minus. And then when you plug in five and for why you get root zero So you you don't subtract anything, so yeah, remember when we do the truths of involving sign We have this restriction on data. Now, the only time Sinan burst could be negative one and this interval. Excuse me. That should've been through there. Only time sign could be negative. One is when state is negative too. So we'LL go ahead and cancel those negatives unless you simplify twenty five. Sign in verse to over five. We could estimate that with the copulate er and let's just go ahead and distribute just twenty five ill So on their next up twenty five's cancel. So we just get a l you know what? Let me actually factor out in l. A. But not a twenty five. Just to simplify this, somebody erased this twenty five over here and now I'm gonna go and distribute that twenty five. So it shows up here on the sign it cancels with this next fraction. And then here I just do plus pirates Who? Okay, so at this point waken just go straight to the calculator on this one and estimate this. So this is approximately fifty eight point seven to three feats where and then here we shouldn't have. We have l in there too. So all times. So the area that this is the units for the area But since we're multiplying by l, it should be a Cuban. So this is the volume of the water. So the fifty eight decimal is coming from the area and then be multiplied by the link. So let's just keep track of this number then. We also have the volume of the tank which is the capacity of the whole tank. So capacity of tank. It's just the volume of the cylinder. And we know the volume of a cylinder Tyr squared age. And in our problem, the H was the language. Your cell in our was five. So we have twenty five times pi times l. So let's go to the next page. Yeah. Hey, so that we had twenty five pi l and if we approximate that with the cock a winner seventy eight point five four l and also typically. So now we just go back to the very beginning and we want to divide the numerator right here. This is the volume of the water over volume of the entire tank. And this is why we didn't need to know the value l Because they cancel here. So we just divide these two numbers and this is approximately using a calculator. And so now this is our percentage. So here, small supplied by hundred. And we couldn't say that the tank is filled two around, say, about seventy four point seventy seven percent capacity. And there's our answer.
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