The region between the graphs of y=x^2 and y=3x is rotated around the line x=3. The volume of the resulting solid is
Oh, we're spinning things around now! Well, I hope you're ready for a wild ride.
To find the volume of the solid that's formed when we rotate the region between the graphs of y=x^2 and y=3x around the line x=3, we'll have to do a bit of math. Don't worry, I'll do the heavy lifting!
First, let's find the points where these two curves intersect. Setting them equal to each other, we get:
x^2 = 3x
Simplifying, we get:
x^2 - 3x = 0
Factoring out an x, we have:
x(x - 3) = 0
So, x = 0 or x = 3.
Now, let's set up the integral to find the volume. We'll use the method of cylindrical shells, which is fancy talk for spinning things around.
The volume will be:
V = 2π ∫[0, 3] (x(3 - x) dx)
Integrating, we get:
V = 2π ∫[0, 3] (3x - x^2) dx
V = 2π [3/2x^2 - 1/3x^3] from 0 to 3
V = 2π (9/2 - 9/3)
V = 2π (9/2 - 3)
V = 9π - 6π
V = 3π
So, the volume of the resulting solid is 3π. And that's no joke, my friend!
To find the volume of the resulting solid when the region between the graphs of y = x^2 and y = 3x is rotated around the line x = 3, you can use the method of cylindrical shells.
Step 1: Sketch the region and the axis of rotation.
Draw the graphs of y = x^2 and y = 3x on a coordinate plane. The region between them represents the shape to be rotated. The line x = 3 is the axis of rotation. Visualizing them will help in understanding the problem better.
Step 2: Determine the limits of integration.
To calculate the volume using cylindrical shells, you need to integrate with respect to x. Find the x-values where the two curves intersect to determine the limits of integration. Set x^2 = 3x and solve for x to find the points of intersection.
x^2 = 3x
x^2 - 3x = 0
x(x - 3) = 0
This gives x = 0 and x = 3 as the points of intersection.
Step 3: Set up the integral.
The formula for the volume using cylindrical shells is V = 2π ∫[a,b] (x(f(x) - g(x))) dx, where f(x) is the upper curve and g(x) is the lower curve.
In this case, f(x) = 3x and g(x) = x^2. The limits of integration are a = 0 and b = 3.
Therefore, the integral becomes V = 2π ∫[0,3] (x(3x - x^2)) dx.
Step 4: Evaluate the integral.
Integrate the function x(3x - x^2) with respect to x using the limits of integration 0 and 3.
V = 2π ∫[0,3] (3x^2 - x^3) dx
The integral yields V = 2π [x^3 - (x^4)/4] evaluated from 0 to 3.
V = 2π [(3^3 - (3^4)/4) - (0 - 0) ]
V = 2π [(27 - 81/4)]
V = 2π [27/4]
V = 27π/2
So, the volume of the resulting solid is 27π/2.
mommy?
the curves intersect at (0,0) and (3,9). So,
v = ∫[0,3] 2πrh dx
where r=3-x and h=3x-x^2
or,
∫[0,9] π(R^2-r^2) dy
where R=(3-y/3) and r=(3-√y)