Find the volume of the solid obtained by rotating the region bounded by
y=5x+25 and 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.
To start, we need to determine the limits of integration. Since the curves intersect at a certain point, we need to find the x-coordinate of this intersection.
Setting the two equations equal to each other, we have:
5x + 25 = 0
Solving for x, we get:
5x = -25
x = -5
So, the region is bounded from x = 0 to x = -5.
Next, we need to express the equations in terms of x instead of y, since we are rotating about the y-axis.
The equation y = 5x + 25 can be rewritten as x = (y - 25) / 5.
Now, we can set up the integral to calculate the volume:
V = ∫[a, b] 2πx * h(x) dx
Where a = -5 and b = 0, h(x) represents the height of the cylindrical shell at each x-coordinate, and 2πx accounts for the circumference of the shell.
The height of the cylindrical shell, h(x), is given by the difference between the upper and lower curves at each x:
h(x) = (5x + 25) - 0
Simplifying this, we have:
h(x) = 5x + 25
Now, we can substitute everything into the volume formula:
V = ∫[-5, 0] 2πx * (5x + 25) dx
Integrating this expression will give us the volume of the solid.