# Finding volume of solid by cross sections?

Find the volume of the solid whose base is the region bounded by 𝑦 = 𝑥² and the line 𝑦 = 1, and whose cross sections perpendicular to the base and parallel to the x-axis are squares.

I'm not sure what the area of the cross section would be. The cross section is a square so it would just be the side length ² but I don't know how to figure out what that side length is. I know that I need to find the area of the cross section and then I can find the volume by doing the integral of the area of cross section

• $y=x^2$ is a parabola, Draw this parabola and then draw a line parallel to $x$-axis to cut it. The distance between the two intersection points (this is called a chord) is the length of the sides of the square of the respective cross-section,
– YNK
Commented Apr 2, 2020 at 8:37

The easiest way to think of this is to recognize that all slices of the volume are squares that are parallel to the $$x-z$$ plane. At any value of $$y$$, the chord will be $$2x=2\sqrt{y}$$. Thus the area of a vertical square is $$A=4y$$. Then the volume is simply
$$V=\int_0^1 A\ dy=4\int_{0}^{1}y\ dy=2$$