How do you find the volume bounded by #y=sqrtx# and the lines y=0 and x=4 revolved about the y=-1?

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↳Redirected from "How do I change #int_0^1int_0^sqrt(1-x^2)int_sqrt(x^2+y^2)^sqrt(2-x^2-y^2)xydzdydx# to cylindrical or spherical coordinates?"

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Answer 1 · 2016-06-23T09:24:43

# (56 pi )/3 #

Desmos

the volume of a small element is given by

#Delta V = pi ((1+sqrt(x))^2 - (1)^2 ) Delta x#
# = pi (1+2 sqrt(x) + x - 1 ) Delta x#
# = pi (2 sqrt(x) + x ) Delta x#

So # V = pi \int_0^4 \ 2 sqrt(x) + x \ dx#

#= pi [4/3 x^{3/2} + x^2/2 ]_0^4#

#= pi( [4/3 * 8 + 8 ]- 0)#

# = (56 pi )/3 #