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 given functions about the y-axis, we can use the method of cylindrical shells. This involves integrating the circumference of the solid at each height (y-value) multiplied by the height of the shell, then summing up these infinitesimally thin cylinders to find the total volume.
First, let's find the points of intersection between the two functions, y = 5x + 25 and y = 0. To do this, set the two equations equal to each other:
5x + 25 = 0
Solving for x, we get:
5x = -25
x = -5
So the region is bounded between x = -5 and x = 0.
Next, let's express the given functions in terms of x to simplify the integration. For the function y = 5x + 25, we can rewrite it as x = (y - 25) / 5.
Now, let's set up the integral. Since we are integrating with respect to y (the height), the integral will go from y = 0 to y = maximum value of y for the region.
The maximum value of y is found by substituting x = 0 into the equation y = 5x + 25:
y = 5(0) + 25 = 25
Therefore, the integral will be:
V = ∫(from 0 to 25) 2πx(y - 0) dy
Substituting x = (y - 25) / 5, we can rewrite the above integral as:
V = ∫(from 0 to 25) 2π((y - 25) / 5)(y - 0) dy
Now, we can simplify and solve this integral to find the volume.