Compute the volume of this solid. please explain step by step

Question

Suppose that the line segment y = x/ 3 + 1, 0 ≤ x ≤ 12, is revolved about the x-axis.

Answer 1

x/3 + 1 is the radius of a circle

integrate pi(x/3+1)^2 dx evaluated from 0 to 12

= pi(x/3+1)^3 with limits 0 to 12

= (pi)(12/3 +1)^3 - (pi)(0+1)^3

= pi(5^3 - 1^3)

= (pi)(125-1)

= 124pi cubic units

to check the solution, look at the 3 dimensional figure on its side, as a cone, or most of the cone. To get the full cone, extend the line y=x/3 +1 to the x axis. it intersects at (-3,0), the top or apex of the cone. Now the full cone has a height of 12+3 = 15, from x=-3 to x=12. Area of its base = pi(5^2) = 25pi

volume of a cone = (1/3)hA = 1/3 of the height times the area of its base = (1/3)15(25pi) = 125pi. But subtract from that the top of the cone which has volume = (1/3)(3)(pi)(1)^2 = pi

125pi = pi = 124pi cubic units

the circle at x=0 has radius=1 and area = pi

the circle at x=12 has radius =5 and area = 25pi