The base of a solid is the region enclosed by the graph of x^2 + 4y^2 = 4 and cross-sections perpendicular to the x-axis are squares. Find the volume of this solid.
a. 8/3
b. 8 pi/3
c. 16/3
d. 32/3
e. 32 pi/3
Thanks in advance! :)
Why did the square cross the x-axis? To find the volume, of course!
To solve this problem, we first need to find the range of values for y. By rearranging the equation, we have:
4y^2 = 4 - x^2
y^2 = (4 - x^2)/4
y = sqrt((4 - x^2)/4)
Now we can calculate the volume using integration, but let's make it more exciting and use the method of disks:
We integrate from x = -2 to x = 2, since those are the values that satisfy the equation of the graph.
The volume of each disk (cross-section) is given by V_disk = (side length)^2 * pi, where the side length is 2y.
So the volume is:
V = integral from -2 to 2 of (2y)^2 * pi dx
V = pi * integral from -2 to 2 of (4 - x^2)/2 dx
V = pi * integral from -2 to 2 of (2 - x^2/2) dx
V = pi * (2x - x^3/6) evaluated from -2 to 2
V = pi * (2(2) - (2^3)/6 - 2(-2) + ((-2)^3)/6)
V = pi * (4 - 8/6 + 4 + 8/6)
V = pi * (16/3)
So the volume of the solid is 16/3 pi.
The correct answer is (c) 16/3!
I hope you had a "square"-tastic time solving this problem!
To find the volume of the solid, we need to integrate the area of the cross-sections with respect to x.
First, let's determine the equation of the ellipse formed by the graph of x^2 + 4y^2 = 4.
Rearranging the equation:
4y^2 = 4 - x^2
y^2 = (4 - x^2)/4
y = ±√((4 - x^2)/4)
y = ±√(4 - x^2)/2
Since perpendicular cross-sections are squares, the side length of each square is given by 2y.
Now, let's integrate the area of each square cross-section.
To find the limits of integration, we need to determine the x-coordinate of the points of intersection of the ellipse and the x-axis.
Setting y = 0:
0 = √(4 - x^2)/2
0 = 4 - x^2
x^2 = 4
x = ±2
So, the limits of integration are x = -2 and x = 2.
Now, we can integrate the area of each square cross-section:
∫[from -2 to 2] (2y)^2 dx
= ∫[from -2 to 2] (2(√(4 - x^2)/2))^2 dx
= ∫[from -2 to 2] (4 - x^2) dx
= ∫[from -2 to 2] 4 dx - ∫[from -2 to 2] x^2 dx
= 4x ∣[from -2 to 2] - (x^3/3) ∣[from -2 to 2]
= 4(2) - 4(-2) - [(2^3)/3 - (-2^3)/3]
= 8 + 8 - (8/3 - 8/3)
= 16
Therefore, the volume of the solid is 16.
However, the answer choices provided are all fractions. So, let's check if the volume needs to be expressed as a fraction.
To do this, let's calculate the area of a square cross-section at a specific x-coordinate, say x = 1:
Area = (2y)^2
= (2√(4 - 1^2)/2)^2
= (2√(3))^2
= (2)(√3)^2
= 4(3)
= 12
Since the area of the square cross-section is 12 when x = 1, the volume of the solid cannot be expressed as a fraction.
Therefore, none of the answer choices is correct.
To find the volume of the solid, we need to integrate the area of each cross-section perpendicular to the x-axis from the base to the top of the solid.
First, let's start by visualizing the solid and understanding its cross-section. The equation x^2 + 4y^2 = 4 represents an ellipse centered at the origin with the major and minor axes on the x and y-axes, respectively. The base of the solid is this region bounded by the ellipse.
Since the cross-sections are squares perpendicular to the x-axis, each square will have one side along the x-axis and its opposite corners on the ellipse. The length of the side of each square can be determined by finding the y-coordinate of points on the ellipse for given x-values.
Next, let's find the y-coordinate for a point on the ellipse for a given x-value. Since x^2 + 4y^2 = 4, we can solve for y:
4y^2 = 4 - x^2
y^2 = (4 - x^2)/4
y = ±√((4 - x^2)/4) = ±√(4 - x^2)/2
Since the volume is symmetric with respect to the x-axis, we only need to consider the positive root, so y = √(4 - x^2)/2.
Now, let's express the volume as an integral:
V = ∫[a,b] A(x) dx
where A(x) is the area of the cross-section at x.
Since each cross-section is a square with side length determined by the y-coordinate, the area of each cross-section is A(x) = [√(4 - x^2)/2]^2 = (4 - x^2)/4.
Therefore, the integral becomes:
V = ∫[a,b] (4 - x^2)/4 dx
To find the limits of integration, we need to find the x-values where the ellipse intersects the x-axis. When y = 0, we can solve for x:
x^2 + 4(0)^2 = 4
x^2 = 4
x = ±2
So the limits of integration are -2 to 2:
V = ∫[-2,2] (4 - x^2)/4 dx
Now, let's solve the integral:
V = ∫[-2,2] (4 - x^2)/4 dx
= (1/4)∫[-2,2] (4 - x^2) dx
= (1/4) [4x - (1/3)x^3] |[-2,2]
= (1/4) [(8 - (8/3)) - (-8 - (8/3))]
= (1/4) [(24/3 - 8/3) - (-24/3 - 8/3)]
= (1/4) [(16/3) - (-32/3)]
= (1/4) [(16/3) + (32/3)]
= (1/4) (48/3)
= (1/4) (16)
= 4/4
= 1
Therefore, the volume of the solid is 1.
However, none of the given answer choices match the calculated volume of 1. Please check the answer choices again or verify the question.
base is ellipse
integral of (2y)^2 dx from x = -2 to x = +2
whic is twice the integral from 0 to +2 of 4 y^2 dx
which is
twice integral from 0 to +2 of (4-x^2) dx
2 * [ 4 x -x^3/3] at x = 2
2 * [ 8 - 8/3)
2 * [16/3] = 32/3