The volume of the solid obtained by rotating the region bounded by y=x^2−2x and y=x about the line y=9, has the form a/bπ, where a and b are positive coprime integers. What is the value of a+b?
The curves intersect at (0,0) and (3,3)
Using discs (washers), the volume is
v = ∫[0,3] π(R^2-r^2) dx
where r=9-x and R = 9-(x^2-2x), so
v = π∫[0,3] (9-(x^2-2x))^2 - (9-x)^2 dx
= π(x^5/5 - x^4 - 5x^2 + 27x^2) [0,3]
= 378/5 π
Using shells, we need to separate the region into two parts, because the parabola has 2 values of x for each value of y.
The two branches of the parabola are
x = 1±√(1+y)
The volume is thus
∫[-1,0] 2πrh dy
where r=9-y and h = (1+√(1+y)) - (1-√(1+y))
+∫[0,3] 2πrh dy
where r=9-y and h = 1+√(1+y)-y
2π∫[-1,0] (9-y)(2√(1+y)) dy
= 8/15 π(47-3y)(1+y)√(1+y)
= 376/15 π
2π∫[0,3] (9-y)(1+√(1+y)-y) dy
= 2/15 π (1+y)(5y^2 - 80y + 215 - 2(3y-47)√(1+y)) [0,3]
= 758/15 π
(376/15 + 758/15)π = 378/5 π
To find the volume of the solid obtained by rotating the given region about the line y=9, we can use the method of cylindrical shells.
Step 1: Determine the limits of integration
To find the limits of integration, we need to determine the points of intersection of the two curves y=x^2−2x and y=x.
Setting the two equations equal to each other:
x^2−2x = x
Rearranging the equation:
x^2 − 3x = 0
Factoring out an x:
x(x − 3) = 0
So the solutions are x = 0 and x = 3. These are the limits of integration.
Step 2: Set up the integral
To set up the integral for the volume, we need to express the radius and height of each cylindrical shell in terms of x.
The radius of each cylindrical shell is the distance from the line of rotation (y=9) to the curve. In this case, the radius is 9 - y.
The height of each cylindrical shell can be expressed as the difference between the two functions y=x^2−2x and y=x. So, the height is (x^2 − 2x) - x.
The volume of each cylindrical shell is given by the formula V = 2πrh, where r is the radius and h is the height.
Step 3: Set up the definite integral
The integral for the volume is:
V = ∫(2π)(9 - y)[(x^2 − 2x) - x] dx
Now we need to set up the integral with the appropriate limits of integration:
V = ∫(2π)(9 - x^2 + 2x - x) dx
V = ∫(2π)(9 - x^2 + x) dx
V = ∫(2π)(x + 9 - x^2) dx
Step 4: Evaluate the integral
Integrating the expression:
V = 2π[(1/2)x^2 + 9x - (1/3)x^3] from x = 0 to x = 3
V = 2π[(1/2)(3)^2 + 9(3) - (1/3)(3)^3] - 2π[(1/2)(0)^2 + 9(0) - (1/3)(0)^3]
V = 2π[(1/2)(9) + 27 - (1/3)(27)] - 2π[(1/2)(0) + 0 - (1/3)(0)]
V = 2π[4.5 + 27 - 9] - 2π[0]
V = 2π[22.5]
Step 5: Simplify the volume and find a and b
The volume obtained is 45π, which can be written in the form a/bπ as 45/1π.
The value of a+b is 45+1= 46.
Therefore, the value of a+b is 46.