The base of a solid S is the simicircular region enclosed by the graph of y = the sqrt of [4-x^2] and the x-axis. If the cross sections of S perpendicular to the x-axis are squares, the the volume of S is...?
I've worked this out a few ways using what I know abt finding the area under a curve, but this problem is multiple choice and I don't come up with any of these:
a) 32pi/3
b)16pi/3
c)40/3
d)32/3
e)16/3
Show your work. I think you are in error.
14/c squared *c small 5/2c
To find the volume of the solid S, we need to integrate the area of each square cross-section as we move along the x-axis.
First, let's determine the equation of the semicircular region. The graph of y = √(4 - x^2) is the top half of a circle with a radius of 2 units. Therefore, the equation of the circle is x^2 + y^2 = 4.
To find the limits of integration, we need to determine the x-values where the semicircle intersects the x-axis. Setting y = 0 in the equation of the circle, we get:
x^2 + 0^2 = 4
x^2 = 4
x = ±2
Since we only consider the right half of the circle (x ≥ 0), the limits of integration are 0 to 2.
Now let's calculate the area of a single square cross-section. The width of each square is dx (the differential width along the x-axis). Since the base of S is a semicircular region, the height of each square is given by the equation of the semicircle, which is y = √(4 - x^2).
Therefore, the area of each cross-section is (side length of the square)^2 = (√(4 - x^2))^2 = 4 - x^2.
To find the volume of S, we integrate the area function over the given integral limits:
V = ∫(0 to 2) 4 - x^2 dx
Evaluating this integral:
V = [4x - (x^3)/3] (from 0 to 2)
V = 4(2) - (2^3)/3 - (4(0) - (0^3)/3)
V = 8 - 8/3
V = 24/3 - 8/3
V = 16/3
Therefore, the volume of solid S is 16/3, which matches option e) in the multiple-choice options.