the base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line y=1-x. if cross sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid?
area of semicircle = (1/2) pi r^2 = (1/2) pi D^2/4 = (pi/8) D^2
I used D instead of r because the solid seems to be made up of circles sitting on this triangle with radius = y/2 and centered at y/2
So cross sectional area =(pi/8)y^2
so dV = (pi/8) dx [ (1-x)^2 ]
integrate from x = 0 to x = 1
(pi/8) [ 1 dx - 2 x dx + x^2 dx ]
=(pi/8) [ x -x^2 + (1/3)x^3 ] at x = 1 because all terms are 0 at x = 0
= (pi/8)(1/3)
= pi/24
Integrate pi(1-x)^2/2 dx from 0 to 1.
(let u = 1-x )
V = Integral pi u^2/2 du from 0 to 1
= pi/6
Each pi (1-x)^2 dx slab is a slab perpendicualr to the x axis
Damon is right. The individual slab area are pi (1-x)^2/8. I confused diameter with radius
To find the volume of the solid, we need to understand its shape.
The base of the solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line y = 1 - x. Let's visually represent this region:
```plaintext
|
x | y = 1 - x
|
|___________________
x-axis
```
The region is a right triangle with its hypotenuse defined by the line y = 1 - x.
Now, let's consider the cross sections of the solid perpendicular to the x-axis. The problem states that these cross sections are semicircles. This means that each cross section is a half-circle positioned on top of the base.
To calculate the volume, we need to integrate the areas of these half-circles across the length of the base.
First, we express the base area in terms of x. The base is a right triangle, so its area equals one-half the product of its base and height:
Area(x) = (1/2) * (x) * (1 - x)
Now, we can find the volume by integrating the area function over the interval of x-values that define the base.
Volume = ∫[a, b] Area(x) dx
To determine the limits of integration, we need to find the x-values where the base ends. From the given information, we know that the base is bounded by the x-axis, y-axis, and the line y = 1 - x. To find these limits of integration, we set y equal to zero and solve for x:
0 = 1 - x
x = 1
Thus, the limits of integration are from x = 0 to x = 1.
Now, we can calculate the volume:
Volume = ∫[0, 1] (1/2) * (x) * (1 - x) dx
Integrating this expression will give us the volume of the solid.