Having trouble figuring these two out:
1. Consider the solid obtained by rotating the region bounded by the given curves about the y-axis.
y^2=2x, x=y
Find the volume of this solid
2. Consider the solid obtained by rotating the region bounded be the given curves about the line y=-1.
y=(2/x), y=0, x=1, x=3
Find the volume of this solid
I can integrate once I get it set up, Im just having trouble getting them set up.
Thanks in advance!
1. Consider the solid obtained by rotating the region bounded by the given curves about the y-axis.
y^2=2x, x=y
Find the volume of this solid
2. Consider the solid obtained by rotating the region bounded be the given curves about the line y=-1.
y=(2/x), y=0, x=1, x=3
Find the volume of this solid
I can integrate once I get it set up, Im just having trouble getting them set up.
Thanks in advance!
-
1.- The curves cut when y^2 = 2y , ie, at y=0 and y=2 , since y=x , they cut at x=0 and x= 2
Shell method , A= 2pi x H , where H (x) = sqrt2x - x .- ( H is Y parabola - Y line at x)
dV = Adx
dV= 2pi x( sqrt 2x-x) dx
V= 2pi INT x( sqrt2x -x) dx
0
V= 2pi INT sqrt2 x^( 3/2) -x^2) dx
V= 2pi ( sqrt2 ( 2/5) x^(5/2) -x^3/3)
V= 2pi ( sqrt2 (2/5) (4sqrt2 -(8/3)
V= 2pi( 16/5 -8/3)
V= 2pi ( 8/15)
V=16pi /15
2.- Disk washers , Radius of revolution R= (2/x) -(-1) = (2/x)+1
V= INT piR^2 dx
V= pi INT (2/x)+1)^2 dx
1
V= pi ( INT (4/x^2 +4/x +1) dx
V= pi ( ( -4)(1/x) + 4 lnx +x)
V= pi (( -4/3+ 4ln3+3) - (-4+4ln1+1))
V=pi( 5/3+4ln3 +3-0)
V=pi ( 4Ln3 +14/3)
Shell method , A= 2pi x H , where H (x) = sqrt2x - x .- ( H is Y parabola - Y line at x)
dV = Adx
dV= 2pi x( sqrt 2x-x) dx
V= 2pi INT x( sqrt2x -x) dx
0
V= 2pi INT sqrt2 x^( 3/2) -x^2) dx
V= 2pi ( sqrt2 ( 2/5) x^(5/2) -x^3/3)
V= 2pi ( sqrt2 (2/5) (4sqrt2 -(8/3)
V= 2pi( 16/5 -8/3)
V= 2pi ( 8/15)
V=16pi /15
2.- Disk washers , Radius of revolution R= (2/x) -(-1) = (2/x)+1
V= INT piR^2 dx
V= pi INT (2/x)+1)^2 dx
1
V= pi ( ( -4)(1/x) + 4 lnx +x)
V= pi (( -4/3+ 4ln3+3) - (-4+4ln1+1))
V=pi( 5/3+4ln3 +3-0)
V=pi ( 4Ln3 +14/3)