Find the volume of the solid obtained by rotating the region bounded by
y=5x+25, y=0
about the y-axis.
The way you stated it, it would not be a closed region.
I will assume it is also bounded by the y-axis, so we are looking at the right-angled triangle in quadrant II
y = 5x + 25
x = (y-25)/5
x^2 = (y^2 - 50y + 625)/25
= y^2 /25 - 2y + 25
Rotating it about the y-axis
V= π∫ x^2 dy from y = 0 to 25
= π∫( y^2/25 - 2y + 25) dy from 0 to 25
= π[ y^3/75 - y^2 + 25y) form 0 to 25
= π(625/3 - 625 + 625 - 0)
= 625π/3
we can check, since the solid is simply a cone with radius 5 and height of 25
V = (1/3) π r^2 h
= (1/3)π(25)(25) = 625π/3
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 first step is to sketch the region bounded by the curves. In this case, the curves are a line and the x-axis. The line intersects the x-axis at y = 0, and has a positive slope of 5. Therefore, it passes through the point (0, 25) and is increasing as x increases. The region is a triangle with its base on the x-axis and its height determined by the line.
To set up the integral for finding the volume using cylindrical shells, we need to express the height of each shell as a function of y. In this case, the height of each shell can be determined by the equation of the line y = 5x + 25:
y = 5x + 25
Rearranging the equation to solve for x:
x = (y - 25) / 5
Now, we can express the height of each shell as a function of y.
Next, we need to determine the limits of integration. Since the region is bounded by y = 0 and y = 5x + 25, the limits of integration for y will be from 0 to the maximum value of y = 5x + 25. To find the maximum value, we set y equal to 0:
0 = 5x + 25
Solving for x, we get:
x = -5
So, the limits of integration for y are from 0 to 5(-5) + 25, which simplifies to 0 to 0.
Now that we have the formula for the height of each shell and the limits of integration, we can set up the integral to find the volume using the formula for the volume of a cylindrical shell:
V = ∫[a,b] 2πrh dy
where a and b are the limits of integration, r is the distance from the axis of rotation (which is the y-axis in this case), and h is the height of each cylindrical shell.
In this case, r is simply the x-value, which is (y - 25)/5.
Therefore, the integral that represents the volume of the solid is:
V = ∫[0,0] 2π(y - 25)/5 * y dy
Since the limits of integration are both 0, the integral evaluates to 0, which means that the volume of the solid obtained by rotating the region bounded by y = 5x + 25 and y = 0 about the y-axis is 0.