Volume of Solid generated by revolving the region bounded by y=x^2 and the line y=1 about the lines:?

Question

the line y=1
the line y=2
the line y=-1

Also for extra challenge find the volume of the solid generated by revolving the triangular region bounded by lines y=2x and the lines y=0 and x=1 about the line x=2

Answer 1

around the line y = 2, use the washer method.

the solid is a bowl cut by a cylinder. volume is volume of the bowl - vol of the cylinder.

so integrate from x = 0 to x = 1 the vol of a disc with width dx and radius 1 + (1-x^2) minus the vol of the disc with width dx and radius 1.

this is int (0 to 1) [pi((2-x^2)^2) - pi(1^2)]
= pi int(0:1) (4 -4x^2 + x^4) -1)
=pi(x^5/5 -4x^3/3 + 3x)eval(0:1)
=pi(3 -4/3 +1/5) = 28/15*pi.

2: around y = 1, this is a simple bowl. integrate from x = 0 to 1 the volume of a washer with radius dx
is
int(0:1) pi * (1-x^2)^2dx
= pi *int(1 - 2x^2 + x^4)dx
=pi(x^5/5 -2x^3/3 + x)eval(0:1)
=pi(1/5 -2/3 + 1) = 8pi/15.

around y = -1, integrate from x = 0 to 1 the volume of the washer radius 2 and width dx minus the volume of the washer radius 1 + x^2 and width dx.

= pi * int(0:1)(2^2-(1 + x^2)^2)dx
= pi * int(0:1)(4-(1 + 2x^2 + x^4))dx
=pi * (-x^5/5 - 2x^3/3 + 3x)eval(0:1)
=pi*(-1/5 -2/3 + 3) = 32pi/15.

now I'll set up the challenge, but you have to do it yourself. you'll integrate from y = 0 to 2 the area of a washer with radius 1 + (1 -y/2) and width dy, minus the volume of a washer with radius 1 and width dy, this is

int(0:2) [pi*(radius of big washer)^2-pi * (radius of washer with radius 1)^2]dy

you oughtta know what you're doing by now, I've done almost the whole thing already

Answer 2

y=(x-0,3)^2aa2cccd54827fae2b924b9b334db92c remedy for x to get bounds 0,3=(x-0,3)^2 x=0,3 So combine with appreciate to x and that's a volumetric concern giving the kind: ?A(x) = ?pi*r^2 with r being your function for this reason: ?pi*((x-0,3)^2)^2 dx from x=0,3 to x=0,3 (rearranged consistent = pi*?((x-0,3)^2)^2 dx [aa2cccd54827fae2b924b9b334db92caa2cccd54827fae2b924b9b334db92caa2cccd54827fae2b924b9b334db92c] Your very final answer = 24aa2cccd54827fae2b924b9b334db92c*pi/5