Find the volume of the solid whose base is the semicircle y= sqrt(1− x^2) where −1≤x≤1, and the cross sections perpendicular to the x -axis are squares.
the base of the square at x is y=√(1-x^2), so its area is 1-x^2. Then, adding up all these thin slices of thickness dx, and taking advantage of symmetry,
v = 2∫[0,1] (1-x^2) dx
Well, well, well, let's calculate the volume of this solid, shall we? Now, since the base is a semicircle, we need to find its radius first. The equation y = sqrt(1 - x^2) represents the top half of a circle centered at the origin with a radius of 1, which means the radius of our semicircle is... you guessed it, one!
Now, let's talk about those cross sections that are perpendicular to the x-axis. You say they're squares, huh? Well, that makes things quite interesting. Since we're dealing with squares, we know that all sides are equal, so let's call the side length of each square "s."
Now, to find the volume of the solid, we need to integrate the cross-sectional areas from -1 to 1. Each cross section is a square, so its area is simply s^2. And since the square is perpendicular to the x-axis, the height is the same as the width, which is the side length "s."
So, the volume of the solid can be calculated as the integral from -1 to 1 of s^2 ds. Now, let's integrate it to see what we get!
∫(s^2) ds from -1 to 1.
Integrating s^2, we get (1/3)s^3, so now we have (1/3)[s^3] from -1 to 1.
Now let's substitute the limits of integration:
(1/3)[(1)^3] - (1/3)[(-1)^3]
Simplifying this further:
(1/3) - (-1/3)
(1/3) + (1/3)
That gives us 2/3, which is our final answer.
So, the volume of the solid with a semicircular base and square cross sections perpendicular to the x-axis is 2/3.
Now, wasn't that square-dulous?
To find the volume of the solid, we can use the method of cross-sections.
First, let's consider a vertical cross-section of the solid perpendicular to the x-axis. Since the cross-sections are squares, the area of each cross-section will be a square. To find the side length of each square, we need to determine the length of the base of each cross-section.
The base of each cross-section is the length of the corresponding segment on the x-axis. Since the base of the solid is a semicircle with a radius of 1 (from -1 to 1), the length of the base of each cross-section will be 2.
Therefore, the area of each cross-section is given by (side length)^2 = (2)^2 = 4.
To find the volume of the solid, we need to integrate the areas of all the cross-sections over the given interval of x.
V = ∫[from -1 to 1] (area) dx
V = ∫[from -1 to 1] 4 dx
Evaluating the integral, we get:
V = 4 ∫[from -1 to 1] dx
V = 4[x] [from -1 to 1]
V = 4(1 - (-1))
V = 4(2)
V = 8
Therefore, the volume of the solid is 8.
To find the volume of the solid, we first need to determine the height of each square cross section. Since the cross sections are perpendicular to the x-axis, the height of each square will be the difference between the y-coordinates of the top and bottom points of the semicircle at corresponding x-values.
Let's break down the process step by step:
1. Determine the interval of x-values within which the semicircle lies.
- Given: −1 ≤ x ≤ 1
- So, the base of the solid lies within the interval [-1, 1].
2. Find the y-coordinate of the top point of the semicircle at a given x-value.
- The equation of the semicircle is y = sqrt(1 - x^2).
- To find the top point of the semicircle at a specific x-value, substitute the x-value into the equation.
For example, for x = -1, the y-coordinate of the top point is sqrt(1 - (-1)^2) = sqrt(1 - 1) = 0.
Similarly, y-coordinate at x = 0 is sqrt(1 - 0^2) = sqrt(1) = 1.
And the y-coordinate at x = 1 is sqrt(1 - 1^2) = sqrt(1 - 1) = 0.
3. Find the y-coordinate of the bottom point of the semicircle at a given x-value.
- The bottom point of the semicircle lies on the x-axis at y = 0.
4. Calculate the height of the square cross section at a given x-value.
- Subtract the y-coordinate of the bottom point from the y-coordinate of the top point.
For example, for x = -1, the height of the square cross section is 0 - 0 = 0.
Similarly, for x = 0, the height of the square cross section is 1 - 0 = 1.
And for x = 1, the height of the square cross section is 0 - 0 = 0.
5. Use the height of each square cross section to calculate its volume.
- The volume of each square is given by V = side^2, where side is the height.
- For example, the volume of the square at x = -1 is V = 0^2 = 0.
The volume of the square at x = 0 is V = 1^2 = 1.
And the volume of the square at x = 1 is V = 0^2 = 0.
6. Finally, find the total volume by integrating the individual square volumes over the interval of the base.
- The volume of the solid is given by V = ∫(0 to 1) [side^2 dx].
- Calculate the integral of side^2 with respect to x from 0 to 1, and you will have the volume of the solid.
I hope this explanation helps you understand the process of finding the volume of the solid with square cross sections!