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A wedge is cut out of a circular cylinder of radius 4 by two

planes. One plane is perpendicular to the axis of the cylinder. The other intersects the first at an angle of $30^{\circ}$ along a

diameter of the cylinder. Find the volume of the wedge.

$$

\frac{128}{3 \sqrt{3}}

$$

Applications of Integration

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Harvey Mudd College

Baylor University

Idaho State University

Boston College

so once again welcome to new problems. This time we're dealing with a cylinder, so assume you have a right Celinda like that. And it just so happens that theme. The cylinder itself has a new axis. It has a center and an axis. So this is this is the axis off the cylinder and has an origin in the middle, which we're going to call a old. And then the cylinder itself has a radius R which goes from the center up until this edge off course, we haven't axis on X axis, um, along the damage off the cylinder. And then also, we do have a Y axis, which is, as you can recall, perpendicular to the X axis. This is the Y axis. And then we have, um, we have a point. We have two points. So we are interested in a in an edge. So this is this is an edge, a right edge, and we're going to discuss why we have the right edge. So the cylinder has two plans. One plan is perpendicular to the axis off the cylinder, and then the other plane is at an angle to the cylinder. So there's a second plane, which is at an angle to the cylinder. And that angle is, uh, body such that if we draw this, we draw this triangle. We're gonna have a triangle right here. And that triangle has specific points. This is point B. Thesis point A on the triangle is special because, uh, it's part overweight. So if you think about it, you almost have ah, ship like that. Okay, we have a shape like that on the bottom off the shape is semicircular. So it goes all the way like that. Onda, there's a surface. There's a surface right there. So this is the wedge. This is the wage that you're seeing that's been developed by two planes. There's a plane going that way are 30 degrees on. Then there's a plane which is going that way, which is at 90 to the axis s. So this is your X axis on. This is your Y axis. So those are the two planes you're looking at? Off course, The top top part. So this is a point. This is point B. This is point A. So this is B. Uh, this is point A, and then this is point to see. We have 33 points. This is a right triangle that's gonna help us determine the volume off the age. So our goal in this particular problem is to determine becoming the volume Mm. Off the wedge are using into girls when the radius off the cylinder is r equals two. The radio off the cylinder is r equals to fall as the radius we're dealing with for this particular cylinder R equals to fall. That's our radius and the angle. Uh, this angle right here, this data, uh, data is 30 degrees. That's the angle. So we want to get the radius who want to get the volume off the wedge, given the angle, all of that we're dealing with, so our main focus will be on the triangle. First of all, we want to get the area of the triangle. This point is a This is B thesis. See, this is 30. This is the height of the triangle. And this is why Because I remember the X axis is coming towards you, and the Y axis is going that way. Eso That's what we're calling. This is why on this is Triangle ABC. This is Triangle ABC. So if the, uh, it's the bottom off the cylinder is semicircular like that. This is your wife. So you you're saying the plan view if this if this is our radius, um, if that's our radius and all radius substance to be full, then this semicircular is gonna be, uh, X squared. Plus y squared was to r squared. So the equation off that cycle is X squared, plus y squared equals 2 16. Okay, that's the equation. So this distance right here that you're seeing from this point to that point, uh, can be explained using this equation and so we can solve for why from that. So we have y squared way, have y squared, Why squared equals 2 16 minus x squared. So then why becomes plus minus radical 16 minus x squared. But since well, since this is a distance, it's a positive distance. You're gonna say, why is the same was positive 16 minus x squared. That's a wire right there. I want to get the height of the strangle. So we just saw for the why value you need the height eso if you have a triangle like that. Since the height. This is why equals to radical 16 minus X squared. This is 30. Then we can say that the tangent off 30. Uh, it was to the opposite off the adjacent. And so, hey, just to be radical 16 minus expert times tangent off 30. But then, if you can recall from the 30 60 90 triangle Hmm. This is one. This is two. This is radical three. So 10 gentle 30 is radical. Three, uh, chain gentle 30 is vertical three. So each us to be, um, hate us to be radical. Three times radical, 16 minus x squared. That's our age. We just wanna make sure that we have it. Right. Uh, remember, he h hates over wine. It's 10. 30. So hey, age is white and 30. And given that given that we had a Y value off radical 16 minus x squared a tangent off 30. Uh, actually, actually, we need to backtrack a little bit. This is this is different. This is different. This is 60. This is 30. So tangent off 30 from this diagram is gonna be, um, one of a radical three opposite of adjacent. So this is one over radical three. So each is radical 16 minus expert, all of a radical three. Uh, you want to get the area of the triangle? A X is the area of the triangle eyes. The same was one half base times height, which is one half the base is why, in the high teens age So this is one half the base is Hmm. Radical 16 minus X squared on the heights is Mm. Radical 16, minus X squared. All of a radical three. So the area of the triangle is the same as one half radical, 16 minus X squared. Wanna square that followed the radical three. So it becomes 16, minus x squared for about, uh to radical. Three. 16 minus x squared over to radical three. Now we want to get the volume. Mm. And to get the volume, think about the wedge. The wedge Looks like that. Hmm. On this is semicircular, semi circular. So this part so little bit hidden. And so we have a Siri's. We have a Siri's off, uh, triangles over Siris of triangles moving along the wage. And so, if you Harvard INF intestinal number of triangles, these old triangles, these air triangles these triangles, He's triangles. So if you have infinite decimal number of triangles, they're gonna be running. They're going to be running along this edge from positive are to they give our with the Y axis that way. And this is your X axis. So we're doing the integral from negative. Are too positive are off the area because each one is as an area with respect to X. So when we plug in our limits of integration, we have 16 minus x squared, all of to radical. Three 16, minus expert all over to radical three. So this is one of the two radical three, Integral. Legally 44 16 minus X squared. Mm. Now we want to do the integral of this. I want to do the interval of this. And so I want to say you want to say one over to radical three 16 x minus, X cubed All over three. Well, negative. 4 to 4. So this becomes one over to radical three. The first one is 16 4 minus four. Cute over three minus. The second one is our 16 84 by this negative four cube all over three on. Then this become one over to radical three. 64 miners, 64/3, um, minus negative. 64. You know that. Minus negative. 64 minus negative. 64 off three. So we want to simplify that. We have one of two radical three, 64 minus 64/3, plus 64 minus 64/3, right? Uh huh. So we have one of the two radical three, 64 64 is gonna give us 1 28 minus 1 28 all over three. Then we use the distributive property. So saying this is 1 28 over to radical three, minus 1. 28/6. Radical three. This gives us, um, we can multiply this by three. And this by three. So, uh, 1 28 and 38 threes. 24 mm. Eight. Creative Force created four. All of a six. Radical three minus 1. 28. That's gonna give us who do. 3 84 minus 1. 28. Uh, this is gonna give us six toys to Mm hmm. Bunch of 65 to so 2. 56. All of a six party called three. 2. 56 all over 6 44 weeks to goes here three times and then to goes here 1 28 because 1 28 times two, uh is 2 56. So your final solution for the volume final solution for the volume becomes 1 28/3, Article three. So once again, we had a problem and in this particular problem, were given a right cylinder. Uh, this is the right cylinder with a wedge. That's the wage. And the word is at an angle the two planes off the wager at an angle off 30 degrees. So we wanted to find the volume. Um, off that wage, find the volume, the volume off the wedge. And so we have to find the cross sectional area and then combine those processional areas into volumes is like you're packing them up. We need the height and the wide dimension. So we first start with the wide dimension because the bases semicircular we can assume that the equation of the circle is x squared. Los y squared equals in 16 and then we solve for that on. Then we once we find the the base, which is why we use that in the tangent off study to find the height and we have the base and the height, we can get the area, the area function, the X and then the area function with the integral between negative R and R, and so when simplify that using the integral we get 1 28 3 radical three. So I hope you enjoy the problem. Feel free to send any questions or comments and have a wonderful day.

California State Polytechnic University, Pomona

Applications of Integration