The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.
y2 − x2 = 4, y = 3; about the x-axis
the curves intersect at (±2,√8), and the vertex is at (0,2)
the area is symmetric about the y-axis.
So, using shells of thickness dy, we have
v = 2∫[2,√8] 2πrh dy
where r = y and h = x
v = 2∫[2,√8] 2πy√(y^2-4) dy = 32π/3
using discs (washers) of thickness dx,
v = 2∫[0,4] π(R^2-r^2) dx
where R = 3 and r = y
2∫[0,4] π(9-(x^2+4)) dx = 44π/3
Hmmm. I have to leave just now -- maybe you can spot my mistake.
I'll be back in a few hours, in case it's still a mystery.
that should be
2∫[0,4] π(8-(x^2+4)) dx = 32π/3
dang! I used the wrong limit for x.
If you see it, fix it and redo the math
okay - I've calmed down now. Here's the real solution.
the curves intersect at (±√5,3), and the vertex is at (0,2)
the area is symmetric about the y-axis.
So, using shells of thickness dy, we have
v = 2∫[2,3] 2πrh dy
where r = y and h = x
v = 2∫[2,3] 2πy√(y^2-4) dy = (20π√5)/3
using discs (washers) of thickness dx,
v = 2∫[0,√5] π(R^2-r^2) dx
where R = 3 and r = y
2∫[0,√5] π(9-(x^2+4)) dx = (20π√5)/3
To find the volume of the solid formed by rotating the region bounded by the curves y^2 - x^2 = 4 and y = 3 about the x-axis, we can use the method of calculus known as the "disk method."
The first step is to sketch the region in order to visualize it properly. The equation y^2 - x^2 = 4 represents a hyperbola centered at the origin, and the equation y = 3 is a horizontal line passing through y = 3. The region of interest is the area enclosed between these two curves.
To proceed, we need to determine the limits of integration, which will give us the range of x-values for which the disks will be created. We can find these limits by solving the equation of the curve for x:
y^2 - x^2 = 4
Substituting y = 3 into the equation, we have:
3^2 - x^2 = 4
9 - x^2 = 4
x^2 = 5
x = ±√(5)
So, the limits of integration will be x = -√(5) and x = √(5).
Now, using the disk method, we can calculate the volume of each infinitesimally small disk and sum them up to get the total volume.
The volume of each disk can be approximated by the formula:
dV = π * r^2 * dx
In this case, the radius of each disk is given by the y-coordinate at each x-value, which is 3 - y, and dx represents the infinitesimal width of each disk.
So, the volume of each disk is given by:
dV = π * (3 - y)^2 * dx
To find the total volume, we integrate this expression with respect to x:
V = ∫[x=-√(5) to x=√(5)] π * (3 - y)^2 * dx
Now, we need to express y in terms of x. From the equation y^2 - x^2 = 4, we can solve for y:
y^2 = x^2 + 4
y = ±√(x^2 + 4)
Since we are rotating about the x-axis, we are only interested in the positive y-values, so we take y = √(x^2 + 4).
Substituting this expression into the volume equation and integrating, we get:
V = ∫[x=-√(5) to x=√(5)] π * (3 - √(x^2 + 4))^2 * dx
Evaluating this integral will give us the volume of the resulting solid.