the region R is bounded by the part of the curve y=(x-2)^1.5 for which 2<x<4 the axis and line x=4. find in terms of pi, the volume of solid

R is rotated through four right angles about the x-axis.

the limits are 2<x<4, they are greater than or equal to 2, and smaller than or equal to 4.

Expert Answers

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The solid being referred to here is formed when the curve y = (x - 2)^(1.5) is rotated about the x-axis and x lies between 2 and 4.

The volume of the solid can be considered as the sum of the volume of a series of cylinders of height dx and radius y.

The volume is: Int[pi*y^2 dx], x = 2 to x = 4

=> Int[ pi* ((x - 2)^1.5)^2 dx], x = 2 to x = 4

=> Int[ pi* (x - 2)^3 dx], x = 2 to x = 4

=> pi* Int[(x - 2)^3 dx], x = 2 to x = 4

=> [pi*(x - 2)^4/4], x = 2 to x = 4

=> (pi/4)*( [(4 - 2)^4 - (2 - 2)^4]

=> (pi/4)*(2^4 - 0)

=> pi*4

The required volume is 4*pi

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