Find the volume of the solid formed when the area bounded by x^2-y^2=9, y=x+5, and the x axis is rotated about x=1.
Looks like a small trianglulish area. Curve intersects x-axis at x = -3.
Line intersects at x = -5
Line intersects curve at (-17/5,8/5)
Using discs,
v = ∫pi(R^2-r^2) dy
where R^2 = (1 - (5-y))^2 = (y-4)^2
r^2 = 9+y^2
v = pi*∫[0,8/5](y-4)^2-(9+y^2) dy
= pi*∫[0,8/5](-8y+7) dy
= pi*(-4y^2+7y)[0,8/5]
= 0.96 pi = 3.016
What a messy question!!!
first find the intersection of the hyperbola and the straight line,
I got (-3.4 , 1.6)
from x^2 - y^2=9 ---> x = ±(9-y^2)^(1/2)
then the inner radius = 1 + √(9-y^2)
and (inner radius)^2 = (1+2√(9-y^2) + 9 - y^2
= 10 - y^2 + 2√(9-y^2)
outer radius = 1 - (y-5) = 6-y
(outer radius)^2 = 36 - 12y + y^2
volume = π∫(36-12y+y^2 - (10 - y^2 + 2√(9-y^2)) dy from 0 to 1.6
= π∫( 26 - 12y + 2y^2 + 2√(9-y^2) ) dy from 0 to 1.6
The messiest part is to integrate √9-y^2)
Let's do that separately.....
(after all that below, I noticed that I used x instead of y, so all that below should be in terms of y )</b?
let x = 3sinθ
dx = 3cosθ dθ
∫ √(9 - x²) dx
= ∫ √(9 - (3sinθ)²) 3cosθ dθ
= 3∫ √(9 - 9sin²θ) cosθ dθ
= 9∫√(1 - sin²θ) cosθ dθ
= 9∫√(cos²θ) cosθ dθ
= 9∫cos²θ dθ
= 9/2*∫(1 + cos(2θ)) dθ
= 9/2*(θ + sin(2θ) / 2 ) + C
= 9/2*(θ + sinθcosθ ) + C
Now:
x = 3sinθ
so
x/3 = sinθ
θ = sin ֿ ¹ (x/3)
θ = cosֿ ¹ (√(9 - x²)/3)
so
= 9/2*(θ + sinθcosθ ) + C
= 9/2*(sin ֿ ¹ (x/3) + sin(sin ֿ ¹ (x/3))cos(cosֿ ¹ (√(9 - x²)/3)) ) + C
= 9/2*(sin ֿ ¹ (x/3) + x/3* √(9 - x²)/3 ) + C
= 9/2*sin ֿ ¹ (x/3) + x√(9 - x²)/2 + C
OK, your turn
Sure hope I did not mess up
looking at your
r^2 = 9+y^2
you are rotating it about the y-axis, not x=1
Ouch! Got me!
To find the volume of the solid formed, we can use the method of cylindrical shells. Here's how you can calculate it step by step:
Step 1: Graph the given equations to visualize the region bounded by the curves.
The equation x^2 - y^2 = 9 represents a hyperbola, and y = x + 5 is the equation of a straight line. Plotting both of these on a graph will help us see what region we are working with.
Step 2: Determine the intersection points.
To find the boundaries of integration, we need to find the intersection points of the two curves. Set the equations equal to each other:
x^2 - y^2 = 9,
y = x + 5.
By substituting the value of y from the second equation into the first equation, we get:
x^2 - (x + 5)^2 = 9.
Expanding and simplifying, we have:
x^2 - (x^2 + 10x + 25) = 9,
x^2 - x^2 - 10x - 25 = 9,
-10x - 34 = 9,
-10x = 43,
x = -43/10.
Substituting this x-value into the equation y = x + 5, we find:
y = (-43/10) + 5 = 7/10.
So one intersection point is (-43/10, 7/10).
Step 3: Set up the integral for the volume.
To calculate the volume using cylindrical shells, we integrate the formula 2πrh * ∆x, where r is the distance from the axis of rotation to the shell, h is the height of the shell, and ∆x is the thickness of the shell.
In this case, since the axis of rotation is x = 1, the distance from the axis to each shell is x - 1.
The height of each shell is given by the difference between the upper curve (y = x + 5) and the lower curve (y = -√(9 + x^2)).
The thickness of each shell is ∆x.
Step 4: Determine the limits of integration.
The limits of integration for this problem will be the x-values at which the two curves intersect, which we found earlier to be -43/10 and the x-value where the hyperbola crosses the x-axis. This can be found by plugging in y = 0 into the equation x^2 - y^2 = 9:
x^2 - 0^2 = 9.
x^2 = 9,
x = ±√9,
x = ±3.
Since we are rotating around x = 1, the limits of integration will be from -43/10 to 3.
Step 5: Set up and solve the integral.
The integral to calculate the volume becomes:
V = ∫(x - 1) * (x + 5 - (-√(9 + x^2))) * ∆x,
where ∆x represents the thickness of each shell.
To evaluate this integral, substitute x + 5 - (-√(9 + x^2)) for y in the expression x - 1. Simplify the expression inside the integral, and then integrate with respect to x over the given limits.
Once you integrate, the result will be the volume of the solid formed when the given area is rotated about x = 1.