The arc of the parabola y=x^2 from (1,1) to (2,4) is rotated about the y-axis. Find the area of the resulting surface.
Well, well, well, we have ourselves a mathematics problem, don't we? Don't worry, I'm here to add a touch of humor to this quadratic quest!
To find the area of a surface created by rotating our parabola, we need some mathematical magic. And by magic, I mean calculus!
We can use integration to find the surface area. Since we are rotating around the y-axis, we need to solve for x in terms of y, because our integration limits depend on y.
The equation of our parabola, y = x^2, rearranged gives us x = √y. Now, let's calculate the surface area (A) using our integration trickery:
A = 2π * ∫[a,b] (x * sqrt(1 + (dy/dx)^2)) dy,
where [a,b] are the limits of our integration. In this case, a = 1 and b = 4, since those are the y-coordinates of our parabola.
Since dy/dx = 2x, let's plug in the values and integrate:
A = 2π * ∫[1,4] (√y * sqrt(1 + 4x^2)) dy.
Integrating that bad boy is no joke, so I'll spare you the details. After getting your hands dirty with the integration, you'll eventually get the area of the surface.
Remember, though, that the answer should be a number and not a punchline. So keep at it and embrace the beauty of math!
To find the area of the surface resulting from rotating the arc of the parabola y = x^2 about the y-axis, we can use the method of cylindrical shells.
The formula to find the area of each cylindrical shell is:
A = 2πrh
where A is the surface area, r is the radius of the shell (which is the x-coordinate of the parabola), and h is the height of the shell (which is the difference in y-values of the two points on the parabola).
Let's proceed step-by-step to calculate the area.
Step 1: Determine the equation of the parabola.
The given parabola is y = x^2.
Step 2: Find the limits of integration.
We need to rotate the arc of the parabola from x = 1 to x = 2. So, the limits of integration will be from 1 to 2.
Step 3: Calculate the radius (r) and height (h) for each cylindrical shell.
The radius of each shell is given by the x-coordinate of the parabola, which is simply x.
The height of each shell is the difference in y-values between the two points on the parabola. The two points are (x, x^2) and (x, x^2).
So, h = (x^2 - x^2) = 0.
Step 4: Calculate the surface area of each shell.
Using the formula: A = 2πrh
A_shell = 2π(0)(x) = 0
Step 5: Integrate the area of all the shells.
The total surface area is the integral of A_shell from x = 1 to x = 2:
A_total = ∫[1 to 2] (0) dx
= 0
So, the area of the resulting surface is 0.
To find the area of the resulting surface when the arc of the parabola y = x^2 is rotated about the y-axis, you can use the method of cylindrical shells. Here's how you can proceed:
1. First, find the equation of the curve when the arc of the parabola from (1,1) to (2,4) is rotated about the y-axis. Since the curve is symmetric with respect to the y-axis, you can consider the positive values of x. The equation of the curve is given by x = √y.
2. Next, find the limits of integration. Since the arc goes from y = 1 to y = 4, the integration limits will be from 1 to 4.
3. Now, consider an infinitesimally thin strip along the y-axis with thickness dy. The height of this strip is the value of x at that particular y-coordinate, which is √y.
4. The surface area of this strip is given by the circumference of the cylindrical shell multiplied by its height. The circumference of the shell is given by 2πr, where r represents the distance from the y-axis to the curve which is x = √y. Therefore, the circumference is 2π√y.
5. The surface area of the strip is then 2π√y * dy.
6. Integrate the surface area over the range of y from 1 to 4. The expression you need to integrate is 2π√y * dy.
7. The integral of 2π√y with respect to y is (4π/3)y^(3/2). Evaluate this integral between y = 1 and y = 4.
∫[1 to 4] 2π√y * dy = (4π/3)(4^(3/2) - 1^(3/2))
8. Simplify the expression to get the final answer for the surface area.
(4π/3)(8 - 1) = 56π/3
Therefore, the area of the resulting surface when the arc of the parabola y = x^2 from (1,1) to (2,4) is rotated about the y-axis is 56π/3 square units.
using discs,
v = ∫[1,4] πr^2 dy
where r = x, so r^2 = y
v = π∫[1,4] y dy
= π (1/2 y^2)[1,4]
= π(8 - 1/2) = 15π/2
using shells, we need to add in the cylinder of radius 1 and height 3, volume 3π, which lies inside the curve
v = 3π + ∫[1,2] 2πrh dx
where r = x, h = 4-y = 4-x^2
v = 3π + 2π∫[1,2] x(4-x^2) dx
= 3π + 2π(2x^2 - 1/4 x^4)[1,2]
= 3π + 2π[(8-4)-(2-1/4)]
= 3π + 2π(4 - 7/4)
= 15π/2