Calculate the volume obtained by rotating around y= -3 the area between the curves
x= y^2 -4
x+y+2=0
I did the Volume as an integral between (-4, 0) of
(x^2 +x - 12 - 6(x+4) ^1/2) dx
The method I used is seeing the solid as a washer(?
Can someone help me to verify if the answer I have is correct
PD= I have another slight question, why in some formulas for the volume of a rotating solid appears 2 Pi?
Thank you!
when you use discs, the volume is base*height = πr^2h
when you use shells, the volume is circumference*height*thickness = 2πrh dy
So, for this problem, using shells of thickness dy, we have
v = ∫[-3,2] 2πrh dy = ∫[-3,2] 2π(y+3)((2-y)-(y^2-4)) dy = 625π/6
using discs of thickness dx is a bit trickier, since at x=0, the upper boundary changes from the parabola to the line.
v = ∫π(R^2-r^2) dx
v = ∫[-4,0] π((3+√(x+4))^2-((3-√(x+4))^2 dx + ∫[0,5] π((3+(2-x))^2-((3-√(x+4))^2) dx
= 64π + 241π/6 = 625π/6
To calculate the volume obtained by rotating around the line y = -3, we can indeed use the washer method. The washer method involves integrating the difference in the outer and inner radii of the washers formed by rotating the area between the curves.
First, let's find the points of intersection between the two curves. We have:
x = y^2 - 4 and x + y + 2 = 0
Solving the second equation for x gives us:
x = -y - 2
Setting the expressions for x equal to each other:
y^2 - 4 = -y - 2
Rearranging and solving for y:
y^2 + y - 2 = 0
Factoring the quadratic equation:
(y + 2)(y - 1) = 0
We get two solutions: y = -2 and y = 1.
Now, let's integrate the volume formula. The outer radius (R) is the distance between the line of revolution (y = -3) and curve 1 (x = y^2 - 4):
R = y^2 - 4 + 3 = y^2 - 1
The inner radius (r) is the distance between the line of revolution and curve 2 (x + y + 2 = 0):
r = -y - 2 + 3 = -y + 1
Now we can calculate the volume using the washer method:
V = ∫[a, b] (π(R^2 - r^2)) dy
where [a, b] represents the interval of y values at the points of intersection.
In this case, the interval is [-2, 1]. Therefore, the volume formula becomes:
V = ∫[-2, 1] π((y^2 - 1)^2 - (-y + 1)^2) dy
Simplifying this expression and calculating the integral will give you the volume of the solid.
Regarding your second question, the appearance of 2π in formulas for volume of rotating solids is due to the fact that we are calculating the volume of a three-dimensional shape by rotating a two-dimensional shape around an axis. The 2π represents a full revolution around the axis, providing the complete solid volume. In essence, 2π is the circumferential length of a full circle.