Consider a solid obtained by rotating the region bounded by y=1/x4, y=0, x=4, x=
ID: 2862074 • Letter: C
Question
Consider a solid obtained by rotating the region bounded by y=1/x4, y=0, x=4, x=9
about the y-axis.
(a) Compute the volume of the solid using the method of washers. Illustrate your method by sketching a diagram of a typical washer. Indicate the curves, axis of revolution, intersection points, and inner radius and outer radius of the washer in your diagram.
(b) Compute the volume of the solid using the method of cylindrical shells. Illustrate your method by sketching a diagram of a typical cylindrical shell. Indicate the curves, axis of revolution, intersection points, and shell height in your diagram.
Explanation / Answer
1)
using the Washer Method:
about x-axis
y = x^4
y = 1
A (x) = ? ( outer radius )^2 - ? ( inner radius )^2
A (x) = ? ( 1 - 0 )^2 - ? ( x^4 - 0 )^2
A (x) = ? ( 1 )^2 - ? ( x^4 )^2
A (x) = ? ( 1 ) - ? ( x^8 )
A (x) = ? ( 1 - x^8 )
1
? ? ( 1 - x^8 ) dx
0
. . . . . .. . .. . .. . .. 1
? ( x - (1/9) * x^9 ) ]
. . .. . .. . .. . .. . .. 0
? [ ( ( 1 - 0 ) - (1/9) * ( 1^9 - 0^9 ) ) ]
? [ 1 - (1/9) * ( 1 - 0 ) ) ]
? [ 1 - (1/9) ]
8?/9
Using shell method:
y = x^4 ---> x = +/- y^(1/4)
height --------> y^(1/4) - 0
radius ====> y
1
? 2? * y * y^(1/4) dy = 8?/9
0
--------------------
2)
using the Washer Method:
about y = -2
y = x^4
y = 1
A (x) = ? ( outer radius )^2 - ? ( inner radius )^2
A (x) = ? ( 1 - -2 )^2 - ? ( x^4 - -2 )^2
A (x) = ? ( 1 + 2 )^2 - ? ( x^4 + 2 )^2
A (x) = ? ( 3 )^2 - ? ( x^8 + 2 * 2x^4 + 4 )
A (x) = ? ( 9 ) - ? ( x^8 + 4x^4 + 4 )
A (x) = ? ( 9 - x^8 - 4x^4 - 4 )
A (x) = ? ( 5 - x^8 - 4x^4 )
1
? ? ( 5 - x^8 - 4x^4 ) dx
0 <---- since we are enclosed by y-axis which is x = 0
. . . . . .. . .. . .. . .. . .. . .. . .. . .1
? ( 5x - (1/9) * x^9 - (4/5) * x^5 ) ]
. . .. . .. . .. . .. . .. . .. . .. . .. . .0
? [ ( 5 * (1 - 0) - (1/9) * ( 1^9 - 0^9 ) - (4/5) * ( 1^5 - 0^5 ) ) ]
? [ 5 - (1/9) - (4/5) ]
184?/45
Using shell method:
y = x^4 ----> x = +/- y^(1/4)
y = 1
height --------> y^(1/4) - 0
radius --------> y + 2
1
? 2? * (y + 2) * ( y^(1/4) ) dy = 184?/45
0
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