How do I calculate the volume of this??

-EquinoX-

What is the volume of the region bounded by y = x^2, y = 1, and the y-axis rotated around the y -axis

mathwonk

define a voklume function, as a function of y, by letting V(y) be that portion of the volume lying below height y. then the derivative of this function i the area of the circular face of this portion of volume, i.e. dV/dy = πr^2 where r = x = sqrt(y), so dV/dy = πy. so you guess a formula for V(y) and then plug in y = 1.

-EquinoX-

I should do this using integral.. I was thinking of slicing this region vertically but then how do I represent this in integrals?? I have to take integrals from -1 to 1 right?

I would evaluate this as the integral from 0 to 1 of (1-squareroot of y)^2 dy

is that right??

HallsofIvy

Why in the world would you slice it vertically? Since it is rotated around the y-axis, slices horizontally will be disks with radius x. Each would have area [itex]\pi x^2= \pi y[/itex] and each infinitesmal disc will have volume [itex]\pi y dy[/itex]. Integrate that.

-EquinoX-

I am sorry that's my mistake. I would slice it horizontally and take the integral from 0 to 1. And shouldn't the radius be 1-square root of y?? Because it's the intersection of y = x^2 and y = x

HallsofIvy

Where did you get y= x from? Your original post was:

Every horizontal "slice" is a circle with center at the y-axis, x= 0, and the end of a radius at [itex]x= \sqrt{y}[/itex]. Of course, the area of the disk is [itex]\pi x^2= \pi y[/itex].