Find the volume of the solid obtained by rotating the region bounded by
y=4x^2,x=1,y=0
about the x-axis.
vol = π∫y^2 dx from 0 to 1
= π∫16x^4 dx
= π[(16/5)x^5] from 0 to 1
= π(16/5 - 0] = 16π/5
Let's call that region "Region McSquared" because it's all about those x-squareds. To find the volume of the solid obtained by rotating Region McSquared about the x-axis, we gotta use some calculus clown magic.
First, let's set up the integral. We want to rotate the region between y = 0 and y = 4x^2 about the x-axis, so we need to integrate with respect to y.
Now, let's find the limits of integration. The curve y = 4x^2 intersects the x-axis at (0,0) and (1,4), so we'll integrate from y = 0 to y = 4.
The volume of the solid is given by the formula V = π∫(R^2 - r^2)dy, where R is the outer radius and r is the inner radius.
Since we're rotating about the x-axis, the outer radius R is equal to x, and the inner radius r is equal to 0. So our integrand becomes V = π∫(x^2 - 0^2)dy.
To express everything in terms of y, let's solve the equation y = 4x^2 for x. Taking the square root of both sides gives us x = √(y/4).
Now we're ready to set up the integral. The volume of the solid is V = π∫(√(y/4))^2 dy, integrated from y = 0 to y = 4.
Simplifying that mess, we get V = π∫(y/4) dy, integrated from y = 0 to y = 4.
Integrating (y/4) with respect to y gives us (1/4)y^2/2, which simplifies to (1/8)y^2.
Now let's plug in the limits of integration. Evaluating the integral at y = 4 and y = 0, we get V = π[(1/8)(4)^2 - (1/8)(0)^2].
Simplifying that, we have V = π(1/8)(16) = 2π.
So the volume of the solid obtained by rotating Region McSquared about the x-axis is 2π cubic units. And remember, math can be funny too, just like a clown!
To find the volume of the solid obtained by rotating the region bounded by the curves y = 4x^2, x = 1, and y = 0 about the x-axis, we can use the method of cylindrical shells.
Step 1: Determine the limits of integration.
Since the curves intersect at x = 1, the limits of integration will be from x = 0 to x = 1.
Step 2: Calculate the height of each cylindrical shell.
The height of each cylindrical shell is given by the difference in y-values between the curves. In this case, the height is given by y = 4x^2 - 0 = 4x^2.
Step 3: Calculate the radius of each cylindrical shell.
The radius of each cylindrical shell is simply the x-value for each point on the curve. In this case, the radius is given by r = x.
Step 4: Calculate the volume of each cylindrical shell.
The volume of each cylindrical shell is given by V = 2πrh, where r is the radius and h is the height. Therefore, the volume of each shell is V = 2πx * 4x^2.
Step 5: Integrate the volume expression to find the total volume.
To find the total volume, we need to integrate the volume expression over the given limits of integration. Therefore, the total volume is given by V = ∫(0 to 1) 2πx * 4x^2 dx.
Step 6: Solve the integral.
Integrating the expression, we get V = 2π * ∫(0 to 1) 4x^3 dx.
V = 2π * [ x^4 ] (0 to 1)
V = 2π * (1^4 - 0^4)
V = 2π * 1
V = 2π cubic units.
Therefore, the volume of the solid obtained by rotating the region bounded by y = 4x^2, x = 1, and y = 0 about the x-axis is 2π cubic units.
To find the volume of the solid obtained by rotating the region bounded by the curves y = 4x^2, x = 1, and y = 0 about the x-axis, we can use the method of cylindrical shells.
The volume of each cylindrical shell is given by the formula:
V = 2πrhΔx
In this case, the radius (r) of each cylindrical shell is x, and the height (h) is the difference between the y-values of the curves y = 4x^2 and y = 0.
Let's break down the steps to find the volume:
1. Determine the limits of integration:
To find the volume, we integrate with respect to x. The region is bounded by the curves x = 1 and y = 0. Therefore, the limits of integration are x = 0 (the lower limit) and x = 1 (the upper limit).
2. Set up the integral:
The integral to calculate the volume is:
V = ∫[0,1] 2πx(4x^2 - 0) dx
3. Simplify the integral:
V = ∫[0,1] 8πx^3 dx
4. Integrate:
Using the power rule, we integrate the integral:
V = 8π ∫[0,1] x^3 dx
The integral of x^3 dx is (1/4)x^4.
Applying the limits of integration, we get:
V = 8π [(1/4)(1)^4 - (1/4)(0)^4] = 8π(1/4) = 2π
Therefore, the volume of the solid obtained by rotating the region bounded by y = 4x^2, x = 1, and y = 0 about the x-axis is 2π cubic units.