Calculating Volume of Rotational Solids Using Integration

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In summary, the conversation discusses finding the volume of a solid formed by rotating a region in the first quadrant, bounded by the curves y=x^2 and y=2x, around the x-axis. The person attempted to solve it using integration but encountered an error. They are advised to find the small piece of volume and correct their approach by using the formula for a washer. The conversation concludes with a reminder to post non-conceptual questions in the appropriate forum.
  • #1
zcabral
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Find the volume of the solid formed by rotating the region inside the first quadrant enclosed by
y=x^2
y=2x
about the x-axis.
So i used intergral 0 to 2 (2x-x^2)^2 dx
this is wat i got...
pi*(((2^5)/5)-(4(2^4)/4)+(4(2^3)/3))
its wrong so wat did i do wrong?:confused:
 
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  • #2
That region will make a washer. Your interval [0,2] looks right.

However it might be beneficial to find a small piece of volume of the actual shape.

V=A*h so an infinitely small piece of volume would be dV = A*dh, or more approately in our case:

dV=A*dx

The area is pi*r^2 since it forms a washer, so [tex]A=\pi (r_1^2-r_2^2)[/tex]

Where r1 and r2 are the distances from the x-axis where r1>r2

The Area portion is what you did wrong, do you see where to go from there?

Next time you should post questions like this (non-conceptual) questions in the homework help.
 
Last edited:

What is the formula for finding the volume of a disk?

The formula for finding the volume of a disk is V = πr²h, where V is the volume, r is the radius of the disk, and h is the height or thickness of the disk.

How do you find the volume of a washer?

To find the volume of a washer, subtract the volume of the smaller circle (inner hole) from the volume of the larger circle (outer edge).

Can the volume of a washer be negative?

No, the volume of a washer cannot be negative. It is a measurement of the space occupied by an object and cannot have a negative value.

What is the difference between a disk and a washer?

A disk is a circular, flat object with a constant thickness, while a washer is a circular object with a hole in the center. The volume of a disk is calculated using its radius and height, while the volume of a washer is calculated by subtracting the volume of the inner hole from the volume of the outer circle.

What are some real-life applications of finding volumes of disks and washers?

Finding volumes of disks and washers is useful in fields such as engineering, architecture, and physics. It can be used to calculate the volume of cylinders, pipes, and other circular objects, as well as in the design of structures and machines. It is also used in calculating the volume of liquids in containers such as tanks and barrels.

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