Question: Find the volume of revolution bounded by y-axis, y=cos(x), and y=sin(x) about the horizontal axis.
Since the rotation is happening at the horizontal axis, I thought the limits of integration would be [-1,1] and if the curves are rotated they would over lap so i used the shell method, thus I got v = ∫[-1,1] 2πy((arccos(y) - arcsin(y)) dy, I didn't get an answer in my calculator. Would doing Disk/Washer method work? I'm not sure if my limits of integration is wrong as well
It is true that both sinx and cosx are bounded by -1 and 1, but the curves intersect at x = π/4
So, the volume, using discs of thickness dx is
v = ∫[0,π/4] π(R^2-r^2) dx
where R=cosx and r=sinx
v = ∫[0,π/4] π(cos^2x-sin^2x) dx
= ∫[0,π/4] πcos(2x) dx = π/2
Using shells of thickness dy, the area has to be split where the curves intersect, so
v = ∫[0,1] 2πrh dy
where r=y and h changes from arcsin(y) to arccos(y) at y = 1/√2
v = ∫[0,1/√2] 2πy*arcsin(y) dy
+ ∫[1/√2,1] 2πy*arccos(y) dy
= π/4 + π/4 = π/2
discs is clearly much easier, since integrating arcsin needs some work.
To find the volume of revolution using the disk/washer method, we first need to determine the limits of integration correctly.
In this case, the region is bounded by the y-axis, y = cos(x), and y = sin(x). We can find the intersection points between these curves by setting them equal to each other:
cos(x) = sin(x)
Dividing both sides by cos(x), we get:
tan(x) = 1
Taking the inverse tangent of both sides, we find the solutions:
x = π/4, 5π/4, 9π/4, ...
The region we want to rotate is between these values of x, in the interval [π/4, 5π/4].
Next, we set up the volume integral using the disk/washer method:
V = ∫[a, b] π(R^2 - r^2) dx
where:
a = π/4 (lower limit of integration)
b = 5π/4 (upper limit of integration)
R = distance between the axis of revolution (horizontal axis) and the outer curve (y = cos(x))
r = distance between the axis of revolution and the inner curve (y = sin(x))
At any given x, R can be found by taking the maximum of the y-values of y = cos(x) and y = sin(x), which is cos(x). Similarly, r can be found by taking the minimum of the y-values, which is sin(x).
Therefore, our volume integral becomes:
V = ∫[π/4, 5π/4] π[(cos(x))^2 - (sin(x))^2] dx
Now you can evaluate this integral to find the volume of revolution.
To find the volume of revolution using the shell method, let's start by understanding the limits of integration.
In this case, the volume is bounded by the y-axis and the curves y = cos(x) and y = sin(x). To determine the limits of integration, we need to find the points where these curves intersect.
Let's equate the two curves to find the intersection points:
cos(x) = sin(x)
We can rewrite this equation as:
cos(x) - sin(x) = 0
To solve this equation, we can use trigonometric identities:
cos(x) - sin(x) = √2(sin(π/4 - x))
Setting this equal to zero, we get:
sin(π/4 - x) = 0
This implies:
π/4 - x = 0
x = π/4
So, the curves intersect at x = π/4.
Since we are rotating around the horizontal axis, the limits of integration should be from y = -1 to y = 1, which is the range of y-values of the curves between the y-axis and the intersection point.
With the correct limits of integration, let's express the volume using the shell method:
V = ∫[-1, 1] 2πy ((arccos(y) - arcsin(y)) dy
Now, to calculate the volume, you can integrate this expression using numerical methods in a calculator, such as a graphing calculator or software like Mathematica.
If you were not able to obtain an answer using the shell method, an alternative approach is using the disk or washer method. In this method, we calculate the volume of each disk or washer and sum them up.
To use the disk method, we need to express the radius of each disk as a function of y. Since the rotation is happening around the horizontal axis, the radius will be the x-coordinate of each curve at a given y-value.
For the curve y = cos(x):
x = arccos(y)
For the curve y = sin(x):
x = arcsin(y)
Applying the disk method, the volume can be calculated as:
V = ∫[-1, 1] π[arccos(y)]^2 - π[arcsin(y)]^2 dy
Again, you would integrate this expression numerically to find the volume.
Both the shell and disk methods should give you the correct volume of revolution. Just be sure to use the proper limits of integration and integrate the expressions correctly.