Finding the volume

Question

A volume is described as follows:

1. the base is the region bounded by y=6−(2/3)x^2 and y=0

2. every cross section parallel to the x-axis is a triangle whose height and base are equal.

Find the volume of this object.

volume = 

Answer 1

Although Yefim S has given the details of the solution, perhaps a sketch might help:

Think of a triangle at some point (like maybe the 2nd triangle up from the x-axis). The volume is the integration of the area of that triangle from y = 0 to y = 6. The area of that triangle is 1/2bh where the base is twice the x-value and the height is the same as that. To know the x-value, you must take the equation of the parabola and switch it around so it is x in terms of y (since you are integrating in the y-direction):

y = 6 - 2/3x²

6 - y = 2/3x²

3/2(6 - y) = x²

9 - 3/2y = x²

x = √(9 - 3/2y)

So the base is double that: 2√(9 - 3/2y) meaning the area is:

A = 1/2[2√(9 - 3/2y)][2√(9 - 3/2y)]

A = 1/2(4)(9 - 3/2y)

A = 2(9 - 3/2y)

A = 18 - 3y or as Yefim used A = 3(6 - y)

Answer 2

Points of integration: 0 ≤ y ≤ 6. Area of section A(y) = 3(6 - y).

Volume V = ∫₀⁶3(6 - y)dy = 3(6y - y²/2)₀⁶ = 3(36 - 36/2) = 54

Answer 3