The base of a solid in the region bounded by the graphs of y = e^-x y = 0, and x = 0, and x = 1. Cross sections of the solid perpendicular to the x-axis are semicircles. What is the volume, in cubic units, of the solid?
the diameter of each semicircle is y. So, adding up all the discs of thickness dx, we have
v = ∫[0,1] 1/2 πr^2 dx
= π/8 ∫[0,1] e^(-2x) dx
and now it's a cinch, right?
To find the volume of the given solid, we first need to determine the shape of the cross sections and then integrate to find the volume.
Given that the cross sections are semicircles, we can visualize the solid as a collection of infinitely many nested semicircles stacked on top of each other along the x-axis.
To find the radius of each semicircle, we observe that the height (y-coordinate) of each semicircle is given by the function y = e^(-x) since it is bounded by the graphs of y = e^-x, y = 0, and the coordinate axes.
Now, we need to find the value of x at each cross section that determines the radius of the semicircle. Let's call this value x_c. This x_c varies from 0 to 1 since we are given that the solid is bounded by x = 0 and x = 1.
Next, we need to express the radius of each semicircle in terms of x_c and find the area of each semicircle using the formula A = (π * r^2)/2, where r represents the radius.
Since the radius is the value of y = e^(-x_c) at each cross section, the area of each semicircle can be expressed as A = (π * (e^(-x_c))^2)/2.
To find the volume, we integrate the area of each semicircle over the interval [0, 1] with respect to x_c, using the formula for volume:
V = ∫₀¹ (π * (e^(-x_c))^2)/2 dx_c
Evaluating this integral will give us the volume of the solid in cubic units.