find the volume of the solid formed by revolving the region bounded by the graph of y=2x^2+4x and y=0 about the y- axis ( express the answernin terms of pie)
To find the volume of the solid formed by revolving the region bounded by the graphs of y = 2x^2 + 4x and y = 0 about the y-axis, we can use the method of cylindrical shells.
First, let's find the points of intersection between the two curves:
Setting y = 2x^2 + 4x = 0, we get:
2x^2 + 4x = 0
x(2x + 4) = 0
So, x = 0 or x = -2/2 = -1.
Next, let's find the limits of integration. Since we are revolving the region about the y-axis, the y-coordinates will determine the limits.
The region is bounded by y = 2x^2 + 4x and y = 0, so the limits of integration will be from y = 0 (the x-axis) to y = 2x^2 + 4x.
Now, we can set up the integral for the volume using cylindrical shells:
V = ∫(2πx)(2x^2 + 4x) dy
The factor of 2πx represents the circumference of the cylindrical shell, and (2x^2 + 4x)dy represents the height or thickness of the shell.
Integrating with respect to y:
V = 2π ∫x(2x^2 + 4x) dy
= 2π ∫(2x^3 + 4x^2) dy
= 2π ∫(2x^3 + 4x^2) dy
Now we substitute the limits of integration:
V = 2π ∫[0, 2x^2 + 4x] (2x^3 + 4x^2) dy
Evaluating the indefinite integral and applying the limits of integration:
V = 2π[x^4/2 + 4x^3/3] |[0, 2x^2 + 4x]
= 2π[(2x^4/2 + 4x^3/3) - (0)]
Simplifying:
V = 2π[(x^4 + 8x^3/3)]
= 2π[(1/3)x^3(x + 8)]
Finally, expressing the answer in terms of π:
V = (2/3)πx^3(x + 8)
That is the volume of the solid formed by revolving the region bounded by the graphs of y = 2x^2 + 4x and y = 0 about the y-axis.