The base of a solid in the xy-plane is the first-quadrant region bounded y = x and y = x2. Cross sections of the solid perpendicular to the x-axis are equilateral triangles. What is the volume, in cubic units, of the solid?
The volume of the solid can be calculated using the formula V = (1/3)Ah, where A is the area of the base and h is the height of the solid.
The area of the base is the area of the first-quadrant region bounded by y = x and y = x2. This can be calculated using the formula A = ∫x2 - x dx, which gives A = (1/3)x3 - (1/2)x2 + c.
The height of the solid is the length of the side of the equilateral triangle, which is equal to 2√3.
Therefore, the volume of the solid is V = (1/3)(1/3)x3 - (1/2)x2 + c)(2√3) = (2/9)x3 - (1/3)x2 + c√3.
To find the volume of the solid, we need to integrate the area of each equilateral triangle cross section along the x-axis.
First, let's consider the shape of the cross section. An equilateral triangle has all three sides equal in length and all three angles equal to 60 degrees. By drawing a diagram, we can see that the triangle's height (h) is equal to the distance between the two given functions, y = x2 and y = x.
To find this distance, we need to find the x-coordinate where the two functions intersect. Setting them equal to each other gives us:
x = x2
Rearranging, we get:
x - x2 = 0
Factoring out an x gives us:
x(1 - x) = 0
So, we have two solutions: x = 0 and x = 1.
Therefore, the height (h) of the equilateral triangle cross section is given by h = x - x2 between x = 0 and x = 1.
Next, let's find the side length of the equilateral triangle. Recall that in an equilateral triangle, each angle is 60 degrees. This means the height divides the base of the triangle into two equal parts, forming a 30-60-90 triangle.
The side length of the equilateral triangle can be obtained using the formula:
s = 2 * (√3/2) * h = √3 * h
Substituting h = x - x2, we get:
s = √3 * (x - x2)
Now, we can find the area of each equilateral triangle cross section:
A(x) = (sqrt(3)/4) * s^2(x)
= (sqrt(3)/4) * (√3 * (x - x2))^2
= (sqrt(3)/4) * 3 * (x - x2)^2
= (sqrt(3)/4) * 3 * (x^2 - 2*x*x2 + x^4)
Finally, to find the volume of the solid, we integrate the area function A(x) with respect to x from x = 0 to x = 1:
V = ∫[0 to 1] (sqrt(3)/4) * 3 * (x^2 - 2*x*x2 + x^4) dx
Evaluating this integral will give us the volume of the solid in cubic units.
To find the volume of the solid, we need to integrate the area of the cross sections along the x-axis.
The region bounded by y = x and y = x^2 in the first quadrant is shown below:
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Since the cross sections are equilateral triangles, the height of each triangle will be the difference between the two curves: x^2 - x.
The area of an equilateral triangle can be calculated using the formula: A = (sqrt(3)/4) * s^2, where 's' is the side length.
In this case, the side length (s) of each triangle will be the distance between the two curves: x^2 - x.
Therefore, the area of each cross section will be: A = (sqrt(3)/4) * (x^2 - x)^2.
To find the volume, we need to integrate this area over the range of x-values that define the region.
The range of x-values is determined by the points of intersection between the two curves, which occur when y = x and y = x^2 are equal. Solving this equation, we get:
x = x^2
x^2 - x = 0
x(x - 1) = 0
So the x-values are 0 and 1. Therefore, the limits of integration are from x = 0 to x = 1.
The volume of the solid can be calculated using the following integral:
V = ∫[0,1] A dx = ∫[0,1] (sqrt(3)/4) * (x^2 - x)^2 dx
Integrating this expression will give us the volume of the solid. Let's solve the integral step by step.
V = (sqrt(3)/4) * ∫[0,1] (x^2 - x)^2 dx
Expanding the square,
V = (sqrt(3)/4) * ∫[0,1] (x^4 - 2x^3 + x^2) dx
Integrating each term,
V = (sqrt(3)/4) * [1/5 * x^5 - 1/2 * x^4 + 1/3 * x^3] evaluated from x = 0 to x = 1
Substituting the limits,
V = (sqrt(3)/4) * [1/5 * 1^5 - 1/2 * 1^4 + 1/3 * 1^3] - [(sqrt(3)/4) * [1/5 * 0^5 - 1/2 * 0^4 + 1/3 * 0^3]
Simplifying,
V = (sqrt(3)/4) * (1/5 - 1/2 + 1/3) - 0
V = (sqrt(3)/4) * (3/30 - 15/30 + 10/30)
V = (sqrt(3)/4) * -2/30
V = (sqrt(3)/4) * -1/15
V = -sqrt(3)/60
Therefore, the volume of the solid in cubic units is -sqrt(3)/60.