Find the exact value of the volume

Question

Suppose that R is the finite region bounded by y=x, y=x+5, x=0, and x=3.

Find the exact value of the volume of the object we obtain when rotating R about the x-axis.

V=

Answer 1

Explanation through step by step:

Given

y=x, y=x+5, x=0, and x = 3.

f(x)=x and g(x)=x+5

a= 0,b=3

To find

Volume=V=?

Formula

The volume of solid of revolution between the curves rotated around the x-axis on an interval [a,b] is given by

V= _a∫^b π|(f(x)² -g(x)²| dx

V=₀∫³ π|(x² -(x+5)²| dx

V= π ₀∫³ |-10x-25| dx

V= π |[(-5x²-25x)] ³ ₀|

V= π|-120|

V=120 π

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Answer 2