The region bounded by y=-x^2+14-45 and y=0 is rotated about the y-axis, find the volume.
The region goes from x=5 to x=9, So,
using shells,
v = ∫[5,9] 2πrh dx
where r = x and h=y
v = 2π∫[5,9] x(-x^2+14x-45) dx
= 448/3 π
Using discs (washers), things get a bit more complicated, because there are two branches to the parabola.
y = 4-(x-7)^2
x = 7±√(4-y)
v = ∫[0,4] π(R^2-r^2) dy
where R = 7+√(4-y) and r = 7-√(4-y)
v = π∫[0,4] (7+√(4-y))^2 - (7-√(4-y))^2) dy
= 448/3 π
Ah, a math question. Time to put on my thinking cap... or should I say my joker cap? Alright, let's dive in!
To find the volume of the region bounded by these two curves when rotated about the y-axis, we're going to use the magical power of integration!
The first step is to find the x-values where these two curves intersect. The region bounded by the curves is the area between the curves from their intersection points to the x-axis. So we need to find the x-values that make y = -x^2 + 14 - 45 = 0.
Solving this quadratic equation brings us to x^2 - 14x + 45 = 0. Now, get ready for a twist! The solutions are x = 5 and x = 9. I guess this quadratic equation just couldn't resist clowning around!
Now that we have the two x-values, it's time to integrate! We need to integrate the equation y = (-x^2 + 14 - 45) - 0 (upper curve - lower curve).
The integral from x = 5 to x = 9 of (-x^2 + 14 - 45) dx gives us the volume of the shape generated by the rotation. Integrating that, we get:
V = ∫[5 to 9] (-x^2 + 14 - 45) dx
Go ahead and plug in those numbers into your favorite integration calculator, because I'm all out of clown tricks for this one!
I hope this helps, and remember to always keep a sense of humor while tackling these math problems. Happy integrating!
To find the volume of the region bounded by the curves y = -x^2 + 14x - 45 and y = 0 when rotated about the y-axis, we can use the method of cylindrical shells.
The first step is to find the intersection points of the curves to determine the limits of integration. Set the two equations equal to each other:
-x^2 + 14x - 45 = 0
Rearrange the equation to find the roots:
x^2 - 14x + 45 = 0
Factoring the equation gives:
(x - 5)(x - 9) = 0
So, x = 5 or x = 9.
Therefore, the limits of integration are from x = 5 to x = 9.
The next step is to determine the height of the cylindrical shell at each point. Since we are rotating the region about the y-axis, the height of the cylindrical shell is the difference between the two curves: h = -x^2 + 14x - 45.
Now, we can set up the integral for the volume using the formula for the volume of a cylindrical shell:
V = ∫[a, b] 2πx * h * dx
V = ∫[5, 9] 2πx * (-x^2 + 14x - 45) dx
Evaluating this integral will give us the volume of the region.
To find the volume of the region bounded by the curves y = -x^2 + 14x - 45 and y = 0 when it is rotated about the y-axis, we can use the method of cylindrical shells.
Cylindrical shells are thin, vertical cylinders that are stacked together to form the shape of the solid. To calculate the volume using cylindrical shells, we need to integrate the circumference of each shell multiplied by its height.
The first step is to find the limits of integration. To do this, we need to find the x-values where the two curves intersect. Set the two equations equal to each other and solve for x:
-x^2 + 14x - 45 = 0
We can factor this quadratic equation:
-(x - 5)(x - 9) = 0
Setting each factor to zero, we find x = 5 and x = 9.
Now we can set up the integral to calculate the volume:
V = ∫[a,b] 2πx( f(x) ) dx
In this case, the limits of integration a and b are 5 and 9 respectively, and f(x) represents the difference between the two curves at a given x-value, which is (-x^2 + 14x - 45) - 0 or simply -x^2 + 14x - 45.
So the integral becomes:
V = ∫[5,9] 2πx( -x^2 + 14x - 45 ) dx
Integrating this expression will give us the volume of the solid formed when the region is rotated about the y-axis. Once the integral is solved, the result will be the volume of the solid.