Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
x+y=4,x=5−(y−1) 2 ;
The curves intersect at (4,0) and (1,3).
You don't specify the axis, so, assuming the x-axis,
using shells, the volume is
v = ∫[0,3] 2πrh dy
where r = y and h = (5-(y-1)^2)-(4-y)
v = 2π∫[0,3] 3y^2 - y^3 dy
= 27π/2
Using washers, we have to break the parabola into its two branches:
y = 1+√(5-x) and y = 1-√(5-x)
v =
∫[1,4] π(R^2-r^2) dx
where R=1+√(5-x) and r = 4-x
+∫[4,5] π(R^2-r^2) dx
where R = 1+√(5-x) and r = 1-√(5-x)
So,
v = ∫[1,4] π((1+√(5-x))^2-(4-x)^2) dx
+ ∫[4,5] π((1+√(5-x))^2-(1-√(5-x))^2) dx
= 65π/6 + 8π/3
= 27π/2
If you meant to revolve around the x-axis, just swap some stuff around.
To find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis, we can use the method of cylindrical shells.
First, let's graph the curves to get a better understanding of the region.
The first curve, x+y=4, is a straight line passing through the points (0, 4) and (4, 0). The second curve, x=5-(y-1)², is a parabola with vertex at (5, 1) and opening downward.
To find the points of intersection between the two curves, we can set them equal to each other and solve for y.
Substituting x=5-(y-1)² into the first equation:
5-(y-1)² + y = 4
5 - (y-1)² + y - 4 = 0
6 - (y-1)² + y = 0
-(y-1)² + y + 5 = 0
This is a quadratic equation in y. Simplifying further, we get:
y² - y - 6 = 0
(y - 3)(y + 2) = 0
So we have two possible values for y: y = 3 and y = -2.
Now, let's consider rotating the region bounded by the curves about the x-axis. The height of each cylindrical shell will be equal to the difference in y-values between the curves at a given x-value.
To integrate, we'll take vertical slices of width dx. The radius of each cylindrical shell will be the x-value itself.
The integral setup for finding the volume V can be written as:
V = ∫[a,b] 2πx * (f(x) - g(x)) dx
where a and b are the x-values where the two curves intersect, f(x) is the curve on top (x = 5-(y-1)²) and g(x) is the curve on the bottom (x + y = 4).
So, to find the volume, we need to calculate the definite integral of 2πx * (f(x) - g(x)) from the x-values where the curves intersect.
Note: In this case, the curves intersect at y = 3 and y = -2. We need to find the corresponding x-values by substituting these y-values into either of the given curves. Once we have the x-values, we can set up the definite integral and evaluate it using appropriate integration techniques.
I hope this explanation helps you understand how to find the volume of the solid obtained by rotating the given region about the specified axis!