How do you find the volume of the wedge-shaped region on the figure contained in the cylinder #x^2 + y^2 = 16# and bounded above by the plane #z = x# and below by the xy-plane?

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Answer 1 · 2018-07-19T13:00:49

# 128/3# cu

#V = int int z dx dy, x >= 0, z in [ 0, x ], and x^2 + y^2 <= 16#

#= int x^2/2 dy, x^2 + y^2 <= 16.#

#= 1/2 int (16- y^2 ) dy. y in [ -4, 4 ]#

#=1/2 [ 16 y - y^3/3 ]#, between y = -4 and y = 4

# = 128/3# cu.