Find the volume of the solid obtained by rotating the region bounded by
y=5x+25 y=0
about the y-axis.
To find the volume of the solid obtained by rotating the region bounded by the curves y = 5x + 25 and y = 0 about the y-axis, we can use the method of cylindrical shells.
The formula for the volume of a cylindrical shell is V = 2πrhΔx, where r represents the distance from the axis of rotation to the edge of the shell, h represents the height of the shell, and Δx represents the thickness of the shell.
In this case, the axis of rotation is the y-axis, and we need to express the equations of the curves in terms of y.
First, let's solve the equation y = 5x + 25 for x:
y - 25 = 5x
x = (y - 25)/5
Now we can set up the integral to find the volume:
V = ∫[a,b] 2πrh dy
To determine the limits of integration (a and b), we need to find the y-coordinate where the two curves intersect. Set them equal to each other and solve for y:
5x + 25 = 0
5x = -25
x = -5
So the curves intersect at y = 0 and y = -5.
Now we can calculate the radius and height of the cylindrical shell at any given y-value:
- For the radius, r, we need to calculate the distance between the y-axis and the curve y = 5x + 25:
r = x = (y - 25)/5
- For the height, h, we need to calculate the difference in y-values for the curves y = 5x + 25 and y = 0:
h = (5x + 25) - 0 = 5x + 25
Now, let's substitute these values into the formula:
V = ∫[-5,0] 2π((y - 25)/5)(5y + 25) dy
Simplifying further:
V = 2π/5 ∫[-5,0] (5y^2 + 25y -125) dy
V = (2π/5) [(5y^3/3) + (25y^2/2) - 125y] ∣ from -5 to 0
V = (2π/5) [-(5(0)^3/3) + (25(0)^2/2) - 125(0) - (5(-5)^3/3) + (25(-5)^2/2) - 125(-5)]
Simplifying further, we find the volume of the solid obtained by rotating the region bounded by the curves y = 5x + 25 and y = 0 about the y-axis.
Please perform the calculations to find the final numerical result.