Find the volumes of the solids generated by revolving the regions bounded by the graphs of the equations about the given lines. y = x2 y = 6x − x2

81π for the x axis.

Step-by-step explanation:

STEP ONE: Determine the intersection.

we are given from the question that y = x^2 and y = 6x − x^2. Therefore if y = x^2, then we will have;

x^2 = 6x - x^2  [1].

Solving and factorizing the equation [1] above give us x = 0 and x = 3 (that is x[6 -2x] = 0 ). Therefore, the point of intersection = (0,0) and (3,9).

STEP TWO: Determine the value for the cross sectional area.

The cross sectional area= [6x - x^2]π - [x2]^2 π. [2].

The cross sectional area = -12 π[x -3]x^2.

STEP THREE: integrate the cross sectional area taking x =3 and x =0 as the upper and lower integration limits or boundaries with respect to dx to determine the vome in the x axis.

volume, v with respect to the x axis = 81π

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