The portion of the ellipse x^2/9+y^2/4=1 with x greater than or equals to 0 is rotated about the y-axis to form a solid S. A hole of radius 1 is drilled through the center of S, along the y-axis. Find the exact volume of the part of S that remains. Show all steps. (Hint: use shells)
Think of the shells as nested cylinders, starting 1 unit away from the y-axis, and extending to the end of the ellipse:
V = ∫[1,3] 2πrh dx
where r=x and h=y
V = 2π∫[1,3] x(2√(1-x^2/9)) dx
= 2π/3 ∫[1,3] 2x√(9-x^2) dx
= 2π/3 (32/3 √2)
= 64/9 √2 π
investigation
To find the volume of the part of solid S that remains after drilling the hole through its center, we can use the method of cylindrical shells.
Step 1: Understanding the problem.
We have an ellipse with the equation x^2/9 + y^2/4 = 1. The portion of the ellipse with x ≥ 0 is rotated about the y-axis to form solid S. A hole with a radius of 1 is drilled through the center of S along the y-axis. We need to find the exact volume of the remaining part of S.
Step 2: Set up the integral.
To use the method of cylindrical shells, we need to consider an infinitesimally small vertical shell. Let's assume the height of the shell is "y". We need to find the volume of this shell.
Consider a small elemental strip on the ellipse at height "y". The length of this strip will be 2πx, and the width will be dy. Since the ellipse equation is given in terms of x and y, we need to express x in terms of y.
From the equation of the ellipse:
x^2/9 + y^2/4 = 1
Multiplying both sides by 9:
x^2 + 9y^2/4 = 9
Rearranging:
x^2 = 9 - 9y^2/4
Taking the square root:
x = √(9 - 9y^2/4)
Step 3: Compute the integral.
Now, we can compute the integral for the volume of the remaining part of S.
The volume of each shell is given by V = 2πrh, where "r" is the distance from the y-axis to the strip, and "h" is the height of the strip.
In this case, r is given by the equation x = √(9 - 9y^2/4), and h is given by dy.
The limits of integration will be between -2 and 2 because the ellipse equation implies that y is between -2 and 2.
Therefore, the integral to compute the volume V is:
V = ∫[from -2 to 2] 2π√(9 - 9y^2/4) dy
Step 4: Calculating the integral.
Let's simplify and solve the integral:
V = 2π∫[from -2 to 2] √(9 - 9y^2/4) dy
= 2π∫[from -2 to 2] √(36/4 - 9y^2/4) dy
= 2π∫[from -2 to 2] √(36 - 9y^2)/2 dy
= π∫[from -2 to 2] √(36 - 9y^2) dy
To solve this integral, we can make a u-substitution. Let:
u = 36 - 9y^2
du = -18y dy
dy = du / (-18y)
Replacing the variables in the integral:
V = π∫[from -2 to 2] √u * (du / (-18y))
= -π/18 ∫[from -2 to 2] √u/√y du
The integration now separates into two parts, one with respect to "u" and the other with respect to "y":
V = -π/18 ∫[from -2 to 2] √u du * ∫[from -2 to 2] 1/√y dy
Simplifying the first integral:
V = -π/18 [u^(3/2) * 2/3] | [from 36 - 9(2)^2 to 36 - 9(-2)^2] * [2√y] | [from -2 to 2]
The values inside the brackets need to be evaluated and then subtracted:
V = -π/18 [(36 - 9(4/3))^(3/2) - (36 - 9(4/3))^(3/2)] * [2√2 - 2√(-2)]
Evaluating the expressions inside the brackets and simplifying:
V = -π/18 [(36 - 9(4/3))^(3/2) - (36 - 9(4/3))^(3/2)] * [2√2 - 2i√2]
Finally, simplify the remaining expression and change the "i" to "√(-1)":
V = -π/18 [(36 - 12)^(3/2) - (36 - 12)^(3/2)] * [2√2 - 2i√2]
= -π/18 [24^(3/2) - 24^(3/2)] * [2√2 - 2i√2]
Since the two terms inside the brackets are the same, they cancel out:
V = -π/18 [0] * [2√2 - 2i√2]
= 0
Therefore, the exact volume of the remaining part of solid S is zero.