Let M be the region under the graph f(x) = 3/(e^x) from x = 0 to x = 5.
M is the base of a solid whose cross sections are semicircles whose diameter lies in the xy plane. The cross sections are perpendicular to the x-axis. Find the volume of this solid.
- I think M is equal to 3 - 3/(e^5)
It does not really matter what the area of M is, but you are correct.
each circle's diameter is a strip under the graph, so adding then all up,
v = ∫1/2 πr^2 dx
where r = y/2 = 3/2 e^-x
v = ∫[0,5]1/2 π(3/2 e^-x)^2 dx = 9π/16 (1 - e^-10)
area of cross section = (1/2) pi (D/2)^2 where D = 3/e^x
= pi D^2 / 8 =(9/8) pi / e^2x
so you want the integral from x = 0 to x = 5 of
(9 pi / 8) e^-2x dx
(9 pi / 8)(-1/2) e^-2x at 5 - at 0
e^-10 is zilch so just - at zero which is +
(9 pi / 8)(1/2)= 9 pi /16
To find the volume of the solid, we need to integrate the area of the cross sections along the x-axis from 0 to 5.
The cross section of each semicircle is a half of a circle, and the radius of each circle is given by the function f(x) = 3/(e^x).
The formula for the area of a circle is A = πr^2, and since we're dealing with semicircles, the formula becomes A = (π/2)r^2.
Let's set up the integral to find the volume:
V = ∫[0, 5] (π/2)(3/(e^x))^2 dx
V = (π/2) ∫[0, 5] (3^2 / (e^x)^2) dx
V = (9π/2) ∫[0, 5] (9 / (e^2x)) dx
V = (81π/2) ∫[0, 5] (1 / (e^2x)) dx
Now, we can integrate the function:
V = (81π/2) ∫[0, 5] e^(-2x) dx
Using the integral rule, we have:
V = (81π/2) * [-1/2e^(-2x)] [0, 5]
Applying the limits of integration:
V = (81π/2) * ( (-1/2e^(-10)) - (-1/2e^(0)) )
V = (81π/2) * ( (-1/2e^(-10)) + 1/2 )
V = (81π/2) * ( 1 - 1/2e^(-10) )
V ≈ 76.937 cubic units
Therefore, the volume of the solid is approximately 76.937 cubic units.
To find the volume of the solid, we can use the method of integration. We need to integrate the area of each cross section along the x-axis.
First, let's find the area of a single cross section. Since the diameter of each semicircle is in the xy-plane, the radius will be half of the height of the function f(x).
The height of the function f(x) is 3/(e^x), so the radius of each semicircle will be (1/2) * (3/(e^x)) = 3/(2*e^x).
The area of a semicircle is given by A = (1/2) * pi * r^2, where r is the radius.
So, the area of each cross section will be A = (1/2) * pi * (3/(2*e^x))^2 = (9*pi)/(8*e^(2x)).
To find the entire volume, we need to integrate this area over the region M.
∫(0 to 5) [(9*pi)/(8*e^(2x))] dx
To evaluate this integral, let's simplify it first.
(9*pi)/(8) is a constant, so we can take it outside the integral.
(9*pi)/(8) * ∫(0 to 5) [1/(e^(2x))] dx
Now, use the integral rule ∫e^x dx = (1/e^x) + C, where C is a constant.
(9*pi)/(8) * [(-1/2)*(1/e^(2x))] (evaluated from 0 to 5)
= (9*pi)/(8) * [(-1/2)*(1/e^(10))] - [(-1/2)*(1/e^(0))]
= (9*pi)/(8) * [(-1/2)*(1/e^(10))] - [(-1/2)*(1/e^(0))]
= (9*pi)/(8) * [(-1/2)*(1/e^(10))] + 1/2
Therefore, the volume of the solid is (9*pi)/(16*e^10) + (pi/16).