Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x+y = 3, x = 4−(y−1)^2; about the
y-axis.
To find the volume of the solid obtained by rotating the region bounded by the curves about the y-axis, we can use the method of cylindrical shells.
Step 1: Determine the limits of integration.
The curves x+y=3 and x=4-(y-1)^2 intersect at two points. We need to find the limits of integration by determining the y-values at these intersection points.
Setting x+y=3 and x=4-(y-1)^2 equal to each other, we get:
4-(y-1)^2 = 3
Simplifying the equation, we have:
(y-1)^2 = 1
Taking the square root of both sides, we get:
y-1 = ±1
y = 2, y = 0
So the limits of integration for y are 0 to 2.
Step 2: Determine the height of each cylindrical shell.
The height of each cylindrical shell is the difference between the y-values of the curves at a given value of y. This can be calculated by subtracting the function values of the curves at that y-value.
For the curve x+y=3, the height (h1) is calculated as follows:
h1 = y
For the curve x=4-(y-1)^2, the height (h2) is calculated as follows:
h2 = (4 - (y-1)^2) - y
Simplifying, we get:
h2 = 4 - (y-1)^2 - y
h2 = 4 - (y^2 - 2y + 1) - y
h2 = 4 - y^2 + 2y - 1 - y
h2 = -y^2 + y + 3
Step 3: Determine the radius of each cylindrical shell.
The radius of each cylindrical shell is the x-value at a given value of y. We can determine the radius by substituting the y-value into one of the curves.
For the curve x+y=3, the radius (r1) is calculated as follows:
r1 = 3 - y
For the curve x=4-(y-1)^2, the radius (r2) is calculated as follows:
r2 = 4 - (y-1)^2
Step 4: Set up the integral for the volume.
The volume of the solid is given by the following integral:
V = ∫(2πy)(r1 - r2)dy
= ∫(2πy)((3 - y) - (4 - (y-1)^2))dy
= ∫(2πy)(-y^2 + y - 1)dy
Step 5: Evaluate the integral.
Integrating the expression, we get:
V = 2π ∫(-y^3 + y^2 - y)dy
= 2π[-(1/4)y^4 + (1/3)y^3 - (1/2)y^2]
Evaluating the integral from y = 0 to y = 2, we get:
V = 2π[-(1/4)(2)^4 + (1/3)(2)^3 - (1/2)(2)^2]
= 2π[-4/4 + 8/3 - 4/2]
= 2π[-1 + 8/3 - 2]
= 2π[-1 + 8/3 - 6/3]
= 2π[-1 + 2/3]
= 2π[1/3]
= (2/3)π
Therefore, the volume of the solid obtained by rotating the region bounded by the curves x+y=3 and x=4-(y-1)^2 about the y-axis is (2/3)π.
The line and the parabola intersect at (3,0) and (0,3)
http://www.wolframalpha.com/input/?i=plot+x%2By+%3D+3,+x+%3D+4%E2%88%92(y%E2%88%921)%5E2
from the straight line: x = 3-y
V = ?? ((4 - (y-1)^2 )^2 - (3-y)^2 ) dy from 0 to 3
= .....
I would expand first, then integrate