The base of a solid consists of the region bounded by the parabola y=rootx, the line x=1 and the x-axis. Each cross section perpendicular to the base and the x-axis is a square. Find the volume of the solid.
Each cross section (from x=0 to x=1) is a square of side √x.
So the total volume is the area of the square times dx, or A(x)dx
V=∫ (√x)^2dx from x=0 to 1
=∫ xdx
=[x²/2] from 0 to 1
= 1/2
To find the volume of the solid, we can use the method of cross-sections. Notice that each cross section perpendicular to the base and the x-axis is a square.
Let's consider a generic cross section at a distance x from the y-axis. The side length of this square cross section will be equal to the height of the parabola at that value of x, which is y = √x.
Since the cross section is a square, its area is (side length)^2 = (√x)^2 = x.
To find the volume of the solid, we need to integrate the areas of all the cross sections from x = 0 to x = 1, which is the range of the base.
So, the volume (V) of the solid can be found by integrating the area of each cross section from x = 0 to x = 1:
V = ∫[0 to 1] x dx
Integrating the function x with respect to x gives:
V = [(1/2)x^2] [0 to 1]
V = (1/2)*(1^2) - (1/2)*(0^2)
V = 1/2
Therefore, the volume of the solid is 1/2 cubic units.