The region in the first quadrant bounded by the x-axis, the line x = ln(π), and the curve y = sin(ex) is rotated about the x-axis. What is the volume of the generated solid?
To find the volume of the solid generated by rotating the region bounded by the x-axis, the line x = ln(π), and the curve y = sin(ex) about the x-axis, we can use the method of cylindrical shells.
The volume of each cylindrical shell can be calculated using the formula:
V = 2π * radius * height * thickness
In this case, the radius of each cylindrical shell will be the distance from the x-axis to the curve y = sin(ex), which is the y-coordinate of the curve. The height of each cylindrical shell can be considered as dx, the thickness of the shell.
To find the limits of integration, we need to determine where the curve y = sin(ex) intersects the x-axis and the line x = ln(π).
To find the x-coordinate where y = sin(ex) intersects the x-axis, we set y equal to 0 and solve for x:
0 = sin(ex)
sin(ex) = 0
=> ex = arcsin(0)
=> ex = nπ, where n is an integer
Since we are considering the region in the first quadrant, we can take n = 0.
ex = 0
=> x = 0
So, the curve y = sin(ex) intersects the x-axis at x = 0.
Now, let's find where the curve y = sin(ex) intersects the line x = ln(π):
x = ln(π)
=> ex = π
=> ln(ex) = ln(π)
=> x = π
So, the curve y = sin(ex) intersects the line x = ln(π) at x = π.
Now we can set up the integral to find the volume:
V = ∫[0 to π] 2π * radius * height * thickness dx
The radius is y = sin(ex), so the radius is sin(ex).
The height or thickness is dx.
Thus, the volume of the generated solid is:
V = ∫[0 to π] 2π * sin(ex) * dx
To evaluate this integral, we can use a table of integrals or numerical methods.
Please note that this explanation assumes basic knowledge of calculus and integration techniques.
integral from x = 0 to x = ln (pi)
of
pi y^2 dx = pi sin^2 e x dx ?
= pi [ x/2 - sin (2 e x)/4e ] at x = ln pi - at x = 0
but 0 at x = 0 so
= pi [ (ln pi) /2 - sin (2 e ln pi)/4e ]
thing is ln pi is just about 1.145 and e is about 2.7183
so 2 e ln pi is about 6.22
which is 2 pi :)
and we know sin 2 pi = 0
so we really have
pi (1.145)/ 2 which is about 1.8