The base of a solid is the region bounded by the parabola y^2=4x and the line x=2. Each plane section is perpendicular to the x-axis is a square. What id the volume of the square?
The volume of the square is 16 units^3.
To find the volume of the solid, we need to understand the shape of the solid formed by the region bounded by the parabola and the line.
First, let's graph the parabola y^2 = 4x and the line x = 2.
The equation y^2 = 4x represents a parabola that opens to the right, with the vertex at the origin (0, 0).
The line x = 2 is a vertical line passing through x = 2.
By graphing these two equations, we can see that the parabola and the line intersect at the point (4, ±2).
Now, we can visualize the solid formed by rotating the region bounded by the parabola and the line around the x-axis.
Since each plane section perpendicular to the x-axis is a square, this means each cross-section of the solid taken perpendicularly to the x-axis will be a square.
To find the volume of each square cross-section, we need to determine the side length of the square.
Let's consider a particular x-value, say x = a.
At this x-value, the corresponding y-value can be found by solving the equation y^2 = 4x for y.
For x = a, we have y = ±√(4a).
Hence, the side length of the square cross-section at x = a is 2√(4a) = 4√a.
To find the volume of the solid, we integrate the area of each square cross-section from x = 0 to x = 2.
The volume V of the solid can be calculated using the following integral:
V = ∫[a=0 to a=2] (side length)^2 da
V = ∫[a=0 to a=2] (4√a)^2 da
V = ∫[a=0 to a=2] 16a da
Evaluating the integral, we get:
V = [8a^2] from 0 to 2
V = 8(2)^2 - 8(0)^2
V = 8(4) - 8(0)
V = 32 cubic units
Therefore, the volume of the square-formed solid is 32 cubic units.
To find the volume of the solid, we need to determine the height of the solid in terms of x and then integrate it.
First, let's solve the equation of the parabola for x in terms of y:
y^2 = 4x
Simplifying, we get:
x = (y^2)/4
Next, we need to find the bounds of integration for x. Since the line x = 2 is the right boundary of the region, the left boundary is the x-coordinate where the parabola and the line intersect.
Setting y^2/4 = 2, we find:
y^2 = 8
y = ±√8 = ±2√2
Now, let's find the height of the solid. Each plane section perpendicular to the x-axis represents a square. The side length of this square is the difference between the y-coordinates of the parabola at the given x-coordinate.
For a given x, the y-coordinate on the parabola is √(4x). Therefore, the height of each square is given by:
h(x) = √(4x) - (-√(4x)) = 2√x
Now, we can integrate the height function from x = 0 to x = 2 to find the volume of the solid:
V = ∫[0 to 2] 2√x dx
Integrating this function, we have:
V = 2∫[0 to 2] x^(1/2) dx = 2(2/3)x^(3/2) |[0 to 2]
V = 2(2/3)(2^(3/2) - 0^(3/2))
V = 2(2/3)(2√2 - 0)
V = 4/3 * 2 * √2
V = 8/3√2
Therefore, the volume of the square solid is 8/3√2.