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Use the Theorem of Pappus to find the volume of the given solid.

A cone with height $ h $ and base radius $ r $.

$\frac{1}{3} \pi r^{2} h$

Applications of Integration

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Campbell University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

problem. We're us to find the volume open. Corn with Brady's are a night eight using purpose. Tear up. Now listen up. This is the X axis. This is the Y axis, and this is Z Axis. Analyst said that we're looking at this column from front of you. So do we have our xx is and why axis and this is or triangle from side view. Saudi Radius is are so this link would be our and height his age This length would be then h All right. Now let's say that we are returning Bess Iran D. Why access? All right now first, what we need is to find the central right off this triangular region and we no, that central off any triangle as theater section of this media. So let's say that we're drawing a meeting starting from the origin, and it's full divine this into equal strips. And according it off it opera point is zero come age. Since this is the origin again in accordance off this point, it's our over zero. So the cordon off this exact point will be our over to an H over too. All right, be noted. Sentra off a triangle is located somewhere along the line and the proportion from the work text up, we're droving medium to As to Kay, Andy, this does between the centrally too deep. This edge is K. It means that in extraction, the difference in coordinates as are over to minus zero. Since this is the origins your zero and that is ICO, tell us saying three K x from this we see that K x is the naked are over six and we're interested in the central sort out Our beat with two k x Annette would be are over feet. So we know that then X bar for the triangle is at our three. Let's say that now we want to find out why bars in my direction the difference between co ordinates s h for two minus zero. And that is because the three k y again s okay. Why would be called H over six and wearing coincident to Okay, Why that beetle toe are Oh, sorry. H over three. All right, so it means that Why borrow said triangle is that Look at it at H over three. So central right off this triangle as central mess off are trying. Go. Is that are over. Three NHL word three. All right, so we have our center and we're located. We are rotating that around. Why access? It means that he would get such a disc. All right, now we first need the circumference off that this is so circumference off that disk would be to pine times the capital R. I will sigh and capital are actually is the, uh, this distance right? And that is our over three feet. So I don't mean to two pi times R b and that is equal to two pi r k b also needy area off the thing triangle that is equal to the multiplication offsides divided by two since that's right, triangle. So eight times are divided by two hr or two again. And the Karen tells us that William is equal to circle friends. Most point by the area and that is equal to two point are over three well supplied by H R. Over too. And that is ICO too. Twos will cancel out b have been pie. Our scribe tends h over three. And this is the answer