1. Find the volume formed by rotating the region enclosed by x=5y and x=y^3 with y is greater than or equal to 0 about the y-axis.
2. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis y=18x-6x^2 , y=0 : about the y-axis.
PLease can anyone help me find the volume plz show ur works if u can so i can understand it better.
1. The volume of the solid obtained by rotating the region bounded by the given curves about the y-axis is given by:
V = 2π ∫0 5y (5y - y3) dy
= 2π ∫0 5y (5y2 - y4) dy
= 2π [5y3/3 - y5/5]|0 5y
= 2π [125/3 - 125/5]
= 250π/15
2. The volume of the solid obtained by rotating the region bounded by the given curves about the y-axis is given by:
V = 2π ∫0 18x-6x2 (18x-6x2) dx
= 2π ∫0 18x2 - 6x4 dx
= 2π [18x3/3 - 6x5/5]|0 18x-6x2
= 2π [54 - 54/5]
= 108π/5
To find the volume formed by rotation, we can use the method of cylindrical shells. Here's how we can approach each problem:
1. For the first question, we have two curves: x = 5y and x = y^3. To find the volume, we need to integrate the area of each cylindrical shell.
First, let's find the limits of integration. Since y is greater than or equal to 0, we can set up the integral to integrate with respect to y.
To find the limits for y, we need to determine where the two curves intersect. Setting them equal to each other, we get:
y = y^3
y^3 - y = 0
y(y^2 - 1) = 0
This equation gives us three possible values: y = 0, y = 1, and y = -1. However, since we are considering y to be greater than or equal to 0, we can ignore the solution y = -1.
So the limits of integration for y will be from 0 to 1.
Now, let's set up the integral. The volume of each cylindrical shell is given by:
dV = 2πr*h*dx
The radius (r) of each shell is given by the x-coordinate of the curve 5y, which is 5y.
The height (h) of each shell is given by the difference in x-values between the two curves, which is x-y^3.
To express everything in terms of y, we need to solve the equations x = 5y and x = y^3 for x.
x = 5y
y = x/5
x = y^3
y^3 = x
y = x^(1/3)
Substituting these values into the expression for the volume, we get:
dV = 2π(5y)(x - y^3)dy
dV = 2π(5y)(y/5 - y^3)dy
dV = 2π(5y^2 - 5y^4)dy
Integrating this expression with respect to y from 0 to 1 will give us the volume:
V = ∫[0,1] 2π(5y^2 - 5y^4)dy
V = 2π ∫[0,1] (5y^2 - 5y^4)dy
Evaluating this integral will give us the final volume.
2. For the second question, we have the curve y = 18x - 6x^2 and the axis y = 0. Following a similar approach, we can set up the integral using cylindrical shells.
First, let's find the limits of integration. Since y = 0 is the axis of rotation, we can integrate with respect to x.
To find the limits for x, we need to determine where the curve intersects the axis y = 0. Setting the equation for the curve equal to 0, we get:
18x - 6x^2 = 0
x(18 - 6x) = 0
This equation gives us two possible solutions: x = 0 and x = 3. So the limits of integration for x will be from 0 to 3.
The expression for the volume of each cylindrical shell is the same as before:
dV = 2πr*h*dx
The radius (r) of each shell is given by the y-coordinate of the curve, which is y = 18x - 6x^2.
The height (h) of each shell is the difference in y-values between the curve and the axis of rotation, which is y - 0 = y.
Substituting these values into the expression for the volume, we get:
dV = 2π(18x - 6x^2)(y)dx
Integrating this expression with respect to x from 0 to 3 will give us the volume:
V = ∫[0,3] 2π(18x - 6x^2)(y)dx
You'll need to express y in terms of x, substitute it into the integral, and evaluate it to find the final volume.
I hope this explanation helps you understand how to find the volume using cylindrical shells! If you have any further questions, feel free to ask.