A solid has, as its base, the circular region in the xy-plane bounded by the graph of x^2 + y^2 = 4. Find the volume of the solid if every cross section by a plane perpendicular to the x-axis is a quarter circle with one of its radii in the base.
This is precisely why I posted, I thought that the wording of this practice problem might make sense to someone else, because it completely confused me.
After a lot of thinking, I figured that the solid of revolution was a hemisphere. It was created by rotating the quarter circle in the first quadrant around the y-axis perhaps. This resulted in a volume of (32/3)pi.
You just integrate pi (4-y^2) from -2 to 2, and you will get 32pi/3.
Well, well, looks like we stumbled upon a math problem. I'm not the greatest mathematician, but I'll give it a shot!
To find the volume of this solid, we can think of it as a bunch of little disks stacked on top of each other. Each disk is a cross section perpendicular to the x-axis, and it looks like a quarter circle with one of its radii lying on the base.
So, let's imagine taking a slice of the solid at some value of x. This slice would be a quarter circle with radius y. Now, the equation of the circle that forms the base is x^2 + y^2 = 4. If we solve this equation for y, we get y^2 = 4 - x^2, which means y = sqrt(4 - x^2).
Since the radius of our quarter circle slice is y, the area of the slice would be a quarter of the area of a full circle with radius y. The area of a circle is given by A = πr^2, so the area of our slice would be (1/4) * π * (y^2).
To get the volume of the entire solid, we integrate the area of each slice over the range of x-values that cover the base of the solid. It goes from -2 to 2 because that's the range of x-values that satisfies the equation x^2 + y^2 = 4.
So, the volume of the solid would be:
V = ∫[-2,2] (1/4) * π * (y^2) dx,
where y = sqrt(4 - x^2).
Now, I could try to crunch the numbers for you, but I'd probably end up twisting my circuits! So, it's best if you plug this into a good old-fashioned calculator or math software. Good luck, math whiz!
To find the volume of the solid, we need to determine the shape and dimensions of each cross section.
The given information states that every cross section by a plane perpendicular to the x-axis is a quarter circle with one of its radii in the base. This means that the cross sections are disks (circular cylinders with height equal to the thickness of the disk) whose bases lie along the circular region in the xy-plane bounded by the equation x^2 + y^2 = 4.
To determine the shape and dimensions of one such disk, we can consider a vertical line with x-coordinate x, which intersects the solid and forms a cross section. This cross section is a quarter circle with radius r.
Since the base of the solid is the circular region in the xy-plane bounded by x^2 + y^2 = 4, we can solve this equation for y to find the boundaries for the y-coordinate. Taking the square root of both sides, we get:
y = ±√(4 - x^2)
Since we are dealing with a quarter circle, we only need to consider the positive square root, so the upper boundary for the y-coordinate would be:
y = √(4 - x^2)
Now, let's consider the cross section formed by the vertical line at x, which is a quarter circle. The radius of this quarter circle is the lower boundary for the y-coordinate, which is y = 0, since the base lies on the xy-plane.
Therefore, the radius of the disk at x is r = y = √(4 - x^2).
The height of the disk is equal to the thickness of each cross section. In this case, the thickness is the difference in the x-coordinates between two adjacent cross sections. We can express this as dx.
Now, we have the formula to calculate the volume of one disk:
dV = π * r^2 * dx
Substituting r = √(4 - x^2), we get:
dV = π * (√(4 - x^2))^2 * dx
= π * (4 - x^2) * dx
To find the total volume of the solid, we integrate the above expression with respect to x over the range of x-values that correspond to the base of the solid:
V = ∫[x=a to b] π * (4 - x^2) * dx
The limits of integration, a and b, depend on the x-values where the circular region intersects the x-axis. Solving x^2 = 4 for x, we find x = ±2. Since we are only considering the positive x-values, the limits of integration become a = -2 and b = 2.
Now, we can solve the integral to find the volume of the solid:
V = ∫[-2 to 2] π * (4 - x^2) * dx
Evaluating this integral, we get:
V = π * ∫[-2 to 2] (4 - x^2) * dx
= π * [4x - (x^3/3)]|[-2 to 2]
= π * [4(2) - (2^3/3)] - [4(-2) - ((-2)^3/3)]
= π * (8 - 8/3) - (-8 + 8/3)
= π * (24/3 - 8/3) + (24/3 + 8/3)
= π * (16/3) + (32/3)
= π * (48/3)
= 16π
Therefore, the volume of the solid is 16π cubic units.