Problem

A general cylinder created by rotating a rectangl…

Problem 282 Medium Difficulty

For the following exercises, use the theorem of Pappus to determine the volume of the shape.
A general cone created by rotating a triangle with vertices (0, 0), (a, 0), and (0, b) around the y -axis.
Does your answer agree with the volume of a cone?

Answer

$\frac{a^{2} b \pi}{3}$

Related Courses

Calculus 1 / AB

Calculus 2 / BC

Calculus

Chapter 2

Applications of Integration

Section 6

Moments and Centers of Mass

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Video Transcript

in the following problem, we went to a German. The volume of the shape generated by rotating the triangle with practices at Sirica Macero, a commissary and serial comma be out from the y axis. And we want to find the volume by its in the theory of steps. So in this case, what goes in the first start off by finding the sketch of this shape So we know it has a purchase. Critics located zero comma zero, another one in Syria committee None. There were not a comma Syria. So it's gonna be the 0.8. Come on, zero. It's when the 0.0 come up. These are not drunk to scale, by the way. So this is a triangle that we want to rotate the bronchi y axis. And if we do so we're gonna find out that we're gonna make a cone. So let me just read, draw these. So we're creating a cone by rotating dis regional here. And so we want to find the volume using the serum of Pappas which states that the volume is equal to the area of the region. That it's been rotated comes a distance that this interview mass travels around this region, So let's first find out the center of Mess. So this interview mass for our wreck tranquil right here with three purchases, has X coordinate of except one plus except two plus except three. You write it by three coma wakes up one plus waist up to was wasted. Three divided by three. If we substitute the valleys over here, we'll have a very big three coma. Be divided by three. That's a recording this for our center of mess. So in this case, this point is going to be located roughly a run here. It will rotate this point around the same axis. We'll find out that he travels around a circle, so the total distance that it travels it's going to be equal did the circumference of the circle. So we know the desert conference can be obtained by multiplying two pi times the radius. In this case, the radius is gonna be the distance from the X axis, and it's located at a coma 38 divided by three units from from the origin. So the distance that it traveled this distance of the center of my struggles. It's gonna be to pie times a divided by three. It's just just simply do pie a bite of a three. Now let's find out their area of the region. This region is a rectangle, so we can actually use the Formula 1/2 the base attempts to height which in this case will have 1/2 times eight, which is the length of its base temps B, which is the length of its height. Just just eight times. Really. But I'd like to. So finally, our volume is given by the area of the region, a temps we divided by two attempts to pi terms. A delighted by three. This ends up being yes, a square times be comes by divided by three. So this is the answer to this question. And before we're done, letters just confirm that this is our answered when other the volume for any cone is just 1/3 of pie. Attempts are squared times height. So over here, the radius is gonna be eight in our high. It's gonna be be we'll have one hat, 1/3 off pie times of radius, which is a square times the height, which is just be. And in this case, we have found out that a squirt him speak time spying divided by three. The same result there. We have obtained using the theory of tapas so we can confirm that this is the correct answer by using geometry.

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