Question Help Use the shell method to find the volume of the solid generated by

Question

Question Help Use the shell method to find the volume of the solid generated by revolving the region bounded by the line y = 5x + 6 and the parabola y =x2 about the following lines. a. The line x= 6 c. The x-axis b. The line x=-1 d. The line y = 36 (a) The volume of the given solid is (Type an exact answer in terms of ?) (b) The volume of the given solid is (Type an exact answer in terms of ?.) (c) The volume of the given solid is ? (Type an exact answer in terms of ?.) (d) The volume of the given solid is (Type an exact answer in terms of ?.)

Explanation / Answer

a)
Using shell method:

height --------> 5x + 6 - x^2
radius ====> 6 - x

limits:
5x + 6 = x^2
0 = x^2 - 5x - 6
0 = (x - 6)(x + 1) ----> x = -1 & 6

6
? 2? * (6 - x) * ( 5x + 6 - x^2 ) dx = 2401?/6
-1

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b)

Using shell method:

height --------> 5x + 6 - x^2
radius ====> x + 1

6
? 2? * (x + 1) * ( 5x + 6 - x^2 ) dx = 2401?/6
-1

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c)
Using shell method:

y = x^2 ----> x = +/- ?(y)
y = 5x + 6 ---> x = (y - 6)/5

height --------> ?(y) - -?(y) -----> 2?(y)
radius --------> y
when x = -1 ------> y = 5(-1) + 6 = 1

height --------> ?(y) - (y - 6)/5
radius --------> y
when x = 6 ------> y = 5(6) + 6 = 36

1. . . .. . . . . . . . . . . . .36
? 2? * y * ( 2?(y) ) dy + ? 2? * y * ( ?(y) - (y - 6)/5 ) dy = 23324?/15
0. . . .. . . . . . . . . . . . .1

--------
d)

Using shell method:

1. . . .. . . . . . . . . . . . . . . . .36
? 2? * (36 - y) * ( 2?(y) ) dy + ? 2? * (36 - y) * ( ?(y) - (y - 6)/5 ) dy = 38416?/15
0. . . .. . . . . . . . . . . . . . . . . 1

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