Question

The base of a solid is the circle x^2 + y^2- 4 . Find the volume of the solid given that the cross sections perpendicular to the x -axis are squares.

a) 134/3
b) 128/3
c) 125/3
d) 137/3
e) 140/3
f) None of the above

Answer 1

To find the volume of the solid with square cross sections perpendicular to the x-axis, you need to integrate the areas of the squares formed by stacking them along the x-axis. The volume can be expressed as V = 8r^2, where r is the radius of the base.

Step-by-step explanation:

To find the volume of the solid, we need to consider that the cross sections perpendicular to the x-axis are squares. If we have a square cross section with side length a, then the area would be a^2. The solid is formed by stacking these squares along the x-axis, so the volume can be found by integrating the areas of the squares.

Using the formula for the area of a circle, we can rewrite the equation of the base as x^2 + y^2 = 4. Solving for y in terms of x, we get y = sqrt(4 - x^2).

Now we can set up the integral to find the volume: V = ∫(0 to 2) (a^2) dx = a^2∫(0 to 2) dx. Evaluating the integral gives V = a^2 * 2 = 2a^2. Since we know a = 2r, we can substitute and get V = 8r^2.

The correct option for the volume of the solid is f) None of the above.