The region in the first quadrant bounded by the x-axis, the line x = π, and the curve y = cos(cos(x)) is rotated about the x-axis. What is the volume of the generated solid?
a. 1.921
b. 3.782
c. 6.040
d. 8.130
think of the volume as a stack of discs of thickness dx, and you have
v = ?[0,?] ?r^2 dx
where r=y=cos(cos(x))
v = ?[0,?] ?cos^2(cosx) dx
This is not an elementary integral, so some numeric method is needed. However, using some symmetry,
v = 2?[0,?/2] ?cos^2(cosx) dx
and you can see that the volume is close to that of a truncated cone.
http://www.wolframalpha.com/input/?i=cos(cos(x))
The curve is close to the line
y = cos(1) + (1-cos(1))/(?/2) x
= 0.54 + 0.29x
So, the peak of the cone, which has been removed has a base radius of 0.54 and a height of 1.862
That makes the semi-volume nearly
?/3 (1^2)(1.862+?/2) - ?/3 (0.54^2)(1.862) = 3.02
So the whole volume is 6.04
This can be checked here:
http://www.wolframalpha.com/input/?i=%E2%88%AB%5B0,%CF%80%5D+%CF%80cos%5E2(cosx)+dx
To find the volume of the generated solid, we can use the method of cylindrical shells.
First, let's find the limits of integration for x. The region is bounded by the x-axis, the line x = π, and the curve y = cos(cos(x)). We need to find the values of x where the curve intersects the x-axis and the line x = π.
Setting y = 0 in the equation y = cos(cos(x)), we have:
0 = cos(cos(x))
To solve this equation, we need to find the values of x where cos(cos(x)) equals 0. This occurs when cos(x) equals π/2 or 3π/2.
The values of x that satisfy cos(x) = π/2 are x = π/3 and x = 5π/3.
The values of x that satisfy cos(x) = 3π/2 are x = 2π/3 and x = 4π/3.
So, the limits of integration for x are π/3 to 2π/3 and 4π/3 to 5π/3.
Now, we can set up the integral to find the volume.
The volume of the generated solid can be calculated using the formula:
V = ∫[a, b] 2πx f(x) dx
Where f(x) is the height of the cylindrical shell at each value of x.
We can express f(x) as the difference between the two curves - the x-axis and y = cos(cos(x)). So,
f(x) = cos(cos(x))
Therefore, the integral for the volume becomes:
V = ∫[π/3, 2π/3] 2πx cos(cos(x)) dx + ∫[4π/3, 5π/3] 2πx cos(cos(x)) dx
To evaluate this integral, we can use numerical methods or a calculator.
After evaluating the integral, the volume of the generated solid is approximately equal to 6.040.
Therefore, the correct answer is c) 6.040.
To find the volume of the generated solid, we can use the method of cylindrical shells.
The region in the first quadrant bounded by the x-axis, the line x = π, and the curve y = cos(cos(x)) can be visualized as a shape between the x-axis and the curve y = cos(cos(x)), and between the x-values 0 and π.
First, let's find the height of each cylindrical shell. The height of each shell will be the difference in the y-coordinates between the curve y = cos(cos(x)) and the x-axis. Since the curve y = cos(cos(x)) is always positive in this region, the height can be written as y = cos(cos(x)).
Next, let's find the radius of each cylindrical shell. The radius will be the distance from the x-axis to the line x = π. This distance is given by the equation r = π - x.
The differential thickness of each cylindrical shell is given by dx, since we are rotating the region about the x-axis.
The volume of each cylindrical shell is given by V = 2πrh*dx, where r is the radius, h is the height, and dx is the thickness.
Integrating V from x = 0 to x = π, we can find the total volume of the generated solid:
V = ∫[0,π] 2π(cos(cos(x)))(π - x) dx
To compute this integral, we can use numerical integration methods or software.
After performing the integration, we obtain the volume of the generated solid to be approximately 3.782.
Therefore, the correct answer is b. 3.782.