The base of a solid is the circle x^2 + y^2 = 9. Cross sections of the solid perpendicular to the x-axis are squares. What is the volume, in cubic units, of the solid?
Awsome...thnx
To find the volume of the solid, we need to integrate the area of the cross sections perpendicular to the x-axis over the range of x-values that the solid occupies.
The given equation x^2 + y^2 = 9 represents a circle with radius 3 centered at the origin. Since the cross sections are squares, every square is equal in area to one side length squared.
To determine the side length of the square at each x-value, we need to find the corresponding y-values based on the equation of the circle. Solving for y in x^2 + y^2 = 9, we get y = ±√(9 - x^2).
Since the cross sections are perpendicular to the x-axis, the side length of the square is equal to 2y. Therefore, the area of each square is (2y)^2 = 4y^2.
Now, we can set up the integral to calculate the volume:
V = ∫[a,b] 4y^2 dx
To determine the limits of integration, we need to find where the circle intersects the x-axis. Setting y = 0 in the equation of the circle, we get x^2 = 9, so x = ±3. Therefore, the limits of integration are -3 and 3 (the x-values where the solid occupies).
V = ∫[-3,3] 4(√(9 - x^2))^2 dx
V = ∫[-3,3] 4(9 - x^2) dx
Now, we can integrate:
V = 4∫[-3,3] (9 - x^2) dx
V = 4 [9x - (x^3/3)] evaluated from -3 to 3
V = 4 [(9*3 - (3^3/3)) - (9*(-3) - ((-3)^3/3))]
V = 4 [(27 - 9) - (-27 - 9)]
V = 4 [18 + 36]
V = 4 * 54
V = 216
Therefore, the volume of the solid is 216 cubic units.
To find the volume of the solid with a circular base and square cross sections, we first need to understand the shape of the solid.
The equation x^2 + y^2 = 9 represents a circle centered at the origin (0,0) with a radius of 3 units. This circle lies in the xy-plane.
Now, let's consider a cross section of the solid perpendicular to the x-axis. This means that the cross section is taken at a specific x-value, and at that x-value, the cross section forms a square.
Since the cross sections are squares, we can determine their side length by considering the length of the chord of the circle at each x-value.
To find the length of the chord, we need to calculate the y-values that lie on the circle for a given x-value.
Let's express the equation of the circle in terms of y to solve for the y-values:
x^2 + y^2 = 9
y^2 = 9 - x^2
y = ± sqrt(9 - x^2)
By taking the square root of (9 - x^2), we get both the positive and negative values of y that correspond to a given x-value on the circle.
For any x-value between -3 and 3, there are two corresponding y-values on the circle, which means there are two points on the circumference of the circle for each x-value.
Now, let's consider the chord formed by these two points on the circle for a given x-value. The length of this chord will be equal to the side length of the square cross section.
Using the distance formula, we can calculate the length of the chord:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For a given x-value, let's say x1, we have two corresponding y-values, y1 and y2.
We can choose either y1 or y2 as the y-coordinate of one of the points, and the other point will automatically be reflected across the x-axis, since the cross section is symmetric.
Therefore, we can calculate the length of the chord using:
d = sqrt((x - x1)^2 + (y1 - -y1)^2)
= sqrt((x - x1)^2 + (y1 + y1)^2)
= sqrt((x - x1)^2 + 4y1^2)
Now, let's find the range of x-values where the circle intersects with the positive y-axis. At y = 0, we have:
0 = 9 - x^2
x^2 = 9
x = ±3
Therefore, the x-values for the positive y-axis intersection are x = 3 and x = -3.
To compute the volume, we integrate the area of each square cross section along the x-axis from x = -3 to x = 3, and then sum them up.
The area of a square cross section is given by side length squared:
A = d^2 = (x - x1)^2 + 4y1^2
To set up the integral, we need to express the side length, d, in terms of x.
Since each cross section is a square, and the side length is equal to d, we can integrate the area as:
V = ∫[(x - x1)^2 + 4y1^2] dx
The limits of integration for x are from -3 to 3, as mentioned before.
Evaluating this integral will give us the volume of the solid.
each square of thickness dx has side 2y, so its area is 4y^2.
Adding up all the thin squares, and using symmetry,
v = 2∫[0,3] 4(9-x^2) dx