Help Finding Volume using Integration...( Disk/washer/Cylindrical shell)?

Question

I've few question that i dont know the answe of so Help please ASAP::::

1)
Consider the solid obtained by rotating the region bounded by the given curve about the line x = 1.
y= x, y=0, x=5, x=6
Find the volume V of this solid?
you can use disk or washer method...

2)
The region bounded by the given curve is rotate about x = -9
x= -1 + y^4, x=0
Find the volume of the resulting solid by any method: ( Disk/washer/Cylindrical shell)


3)
The region bounded by the given curves is rotated about y=7
x=(y-8)^2, x=1
Find the Volume V of the resulting solid by ant method..( Disk/washer/Cylindrical shell)

Thanks

Mathmom · Accepted Answer

1)

Cylindrical Shell Method:

height = x - 0 = x
radius = x - 1
limits: x = 5 to 6

V = 2 π ∫₅⁶ (x-1) x dx
V = 2 π ∫₅⁶ (x² - x) dx
V = 2 π (x³/3 - x²/2) |₅⁶
V = 2 π ((72 - 18) - (125/3 - 25/2))
V = 2 π (54 - 175/6)
V = 2 π (149/6)
V = 149π/3

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Washer method:

Outer radius = 6 - 1 = 5
Inner radius = 5 - 1 = 4
Limits: y = 0 to 5

Outer radius = 6 - 1 = 5
Inner radius = y - 1
Limits: y = 5 to 6

V = π ∫₀⁵ (5² - 4²) dy + π ∫₅⁶ (5² - (y-1)²) dy
V = π ∫₀⁵ 9 dy + π ∫₅⁶ (-y² + 2y + 24) dy
V = π (9y) |₀⁵  + π (-y³/3 + y² + 24y) |₅⁶ 
V = π (45-0)  + π ((-72 + 36 + 144) - (-125/3 + 25 + 120))
V = 45π  + π (108 + 125/3 - 145)
V = 45π  + π (125/3 - 37)
V = 45π  + π (14/3)
V = 149π/3


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2.

Washer method:

Outer radius = 0 - (-9) = 9
Inner radius = -1 + y⁴ - (-9) = y⁴ + 8
Limits: y = -1 to 1 (this is where curve x = -1 + y⁴ intercepts line x = 0)

V = π ∫₋₁¹ (9² - (y⁴ + 8)²) dy
V = π ∫₋₁¹ (-y⁸ - 16y⁴ + 17) dy
V = π (-y⁹/9 - 16y⁵/5 + 17y) |₋₁¹
V = π ((-1/9 - 16/5 + 17) - (1/9 +16/5 - 17))
V = π ((616/45) - (-616/45))
V = 1232π/45

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Cylindrical shell method:

y = ± (x + 1)^(1/4)

Radius = x - (-9) = x + 9
Height = (x + 1)^(1/4) - -(x + 1)^(1/4) = 2(x + 1)^(1/4)
Limits: x = -1 to 0

V = 2π ∫₋₁⁰ 2 (x+9) (x+1)^(1/4) dx
V = 1232π/45

I calculated this last value using WolframAlpha. This is more of a check, to make sure both methods give same result.
http://www.wolframalpha.com/input/?i=integrate+2%CF%80+*2+%28x%2B9%29+%28x%2B1%29^%281%2F4%29+dx%2C+x+%3D+-1+to+0


=========================

3.

Cylindrical Shell Method

Curve x = (y-8)² and line x = 1 intersect at y = 7 and y = 9

Radius = y - 7
Height = 1 - (y-8)² = -y² + 16y - 63
Limits: y = 7 to 9

V = 2π ∫₇⁹ (y-7) (-y² + 16y - 63) dy
V = 2π ∫₇⁹(-y³ + 23y² - 175y + 441) dy
V = 2π (-y⁴/4 + 23y³/3 - 175y²/2 + 441y) |₇⁹
V = 2π ((-6561/4 + 2401/4) + (16767/3 - 7889/3) + (-14175/2 + 8575/2) + (3969-3087))
V = 2π (-1040 + 8878/3 - 2800 + 882)
V = 2π (8878/3 - 2958)
V = 2π (4/3)
V = 8π/3

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Washer method:

y = 8 ± √x

Outer radius = 8 + √x - 7 = 1 + √x
Inner radius = 8 - √x - 7 = 1 - √x
Limits: x = 0 to 1

V = π ∫₀¹ ((1 + √x)² - (1 - √x)²) dx
V = π ∫₀¹ ((1 + 2√x + x) - (1 - 2√x + x)) dx
V = π ∫₀¹ 4√x dx
V = 4π x^(3/2) * 2/3 |₀¹
V = 8/3 π x^(3/2) |₀¹
V = 8/3 π (1 - 0)
V = 8π/3

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Help Finding Volume using Integration...( Disk/washer/Cylindrical shell)?

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