Use the shell method to find the volume of the solid generated by revolving the

Question

Use the shell method to find the volume of the solid generated by revolving the region bounded by the line y = 3x + 4 and he parabola y =x2 about the for following lines. The line x =-1 The line = 16 The line x=4 The x-axis The volume of the given solid is .(Type an exact answer in terms of mu.) The volume of the given solid is . (Type an exact answer in terms of mu. The volume of the given solid is . (Type an exact answer in terms of mu.) The volume of the given solid is . (Type an exact answer in terms of mu.)

Explanation / Answer

y=x^2
y=3x+4
x^2=3x+4
x^2-3x-4=0
x^2-4x+x-4=0
x(x-4)+1(x-2)=0
(x-4)(x+1)=0
x=-1 ; x=4

You integrate from -1 to 4

a)
Since on the interval (-1,4) x+4 > x^2
Let f(x)=x+4-x^2

height = x+4-x^2
radius = 4-x
Volume about the line x=4
= 2 ? ?(4-x)(x+4-x^2) dx --- integrate from -1 to 4

b)
Volume about the line x=-1
= 2 ? ?(x+1)(x+4-x^2) dx --- integrate from -1 to 4

c)
the x-axis : y=0
x=sqrt(y)
x=y-2

Volume = 2 ? ?y (sqrt(y)-y+4) --- integrate from 0 to 16

d)
Volume = 2 ? ?(4-y) (sqrt(y)-y+4) --- integrate from 0 to 16

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