Volume of solid of revolution the rotation curve:
x^2 + y^2 = 25 around the y-axis
Outcome=500*PI/3
I have to be calculated using definite integral. But I don't really know how to calculate it. Can someone help me please?
Volume of solid of revolution the rotation curve:
x^2 + y^2 = 25 around the y-axis
Outcome=500*PI/3
I have to be calculated using definite integral. But I don't really know how to calculate it. Can someone help me please?
You'll need a triple integral, and in my opinion, it'd be easiest to do this using spherical coordinates. If any of this is confusing, let me know and I can simplify it using only rectangular coordinates.
You'll have a triple integral in the order of d(rho)d(phi)d(theta). The limits on theta will be 0 to 2pi (because it's a sphere), the limits on phi will be 0 to pi/2 (the angle it forms with the z axis), and the limits on rho will be 0 to 5 (the distance from the origin to the curve). In the integrand will be 1, since you're finding volume. Understand, or do I need to use rectangular coordinates?
That is, of course, a sphere with radius 5 so you could use the formula as a check.
There are many ways to integrate this. The simplest is probably to use "disks" perpendicular to the y-axis. Solving for x, in the first quadrant so a disk at a specific y would have radius from x= 0 to . The area of such a disk is . Imagining it to have thickness (since the thickness is measured along the y-axis), its volume would be . The volume of all of those disks is the sum which, in the limit as goes to 0, becomes the integral .
Another, slightly harder method would be to use "shells". Imagine the sphere made up of many small cylinders with axis along tye y-axis and radius "x". Each such cylinder would have radius x and height (2 because it goes both up to and down to . Imagining that flattened out to a rectangle with length , the circumference of the cylinder, and height , it would have an area of . Taking the thickness of that cylinder to be (since the thickness is now measured along the y-axis), the volume of such a cylinder would be and the volume of all the cylinders together would be . Taking the limit as goes to 0, that becomes the integral .
Try integrating both ways and see if you don't get . Do you see why the first integral is from y= -5 to 5 and the second from x= 0 to 5?
. It's a central circle with radius . Rotating it around the y-axis would give a sphere.
Volume of the solid of revolution generated with a curve revolving around the y-axis can be calculated using the following expression
Rotating only the quarter of a circle (one in the first quadrant)
would give you one half of the solid of revolution (and half of the volume needed), therefore you should multiply the result by 2.