# Using the definition of the definite integral find the volume of the solid generated   by revolving around the x-axis the triangle with vertices (0,0), (1,0), (1,1):

Asked on by rugby4life

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the volume of solid generated byrotating the triangle found between the function `f(x) = ax + b, x`  axis and limits `x = 0`  and `x = 1` , hence, you need to find first the equationof function f(x) whose graph passes through the points `(0,0)` ) and `(1,1)`  such that:

`f(0) = 0 => 0 = b`

`f(1) = 1 => a*1 + b =1 => a = 1`

Substituting 1 for a and 0 for b in equation of the function f(x) yields `f(x) = x` .

Hence, you need to determine the volume generated rotating the triangle around x axis, such that:

`V = pi*int_0^1 f^2(x) dx => V = pi*int_0^1 x^2 dx`

`V = pi*x^3/3|_0^1`

Using the fundamental theorem of calculus yields:

`V = pi*(1^3/3 - 0^3/3) => V = pi/3`

Hence, evaluating the volume of solid generated by rotating the given triangle about x axis, yields `V = pi/3` .