The base of a solid in the xy-plane is the first-quadrant region bounded y = x and y = x^2. Cross sections of the solid perpendicular to the x-axis are equilateral triangles. What is the volume, in cubic units, of the solid?
So I got 1/30
because (integral from 0 to 1) (x-x^2)^2=1/30
Not quite. You have figured the volume consisting of squares. For the triangle, the base is x-x^2, so the altitude is (x-x^2)√3/2. The volume is then
∫[0,1] (1/2)(x-x^2)(x-x^2)√3/2 dx
= ∫[0,1] √3/4 (x-x^2)^2 dx = √3/120
√3/4 (x^5/5 - x^4/2 + x^3/3) [0,1]
= √3/4 (1/5 - 1/2 + 1/3)
= √3/4 * 1/30
= √3/120
Oh thank you!!!
To find the volume of the solid, we can use the method of slices. Since the cross sections perpendicular to the x-axis are equilateral triangles, we know that the height of each triangle is equal to the length of the base.
Let's break down the problem step by step:
1. Find the length of the base of each equilateral triangle at a given value of x:
The base of each triangle is the difference between the y-coordinate of the upper function (y = x^2) and the lower function (y = x). So, the base length can be expressed as: base = x^2 - x.
2. Find the area of each equilateral triangle at a given value of x:
We know that the area of an equilateral triangle can be calculated using the formula: area = (sqrt(3)/4) * side^2, where side is the length of one of the sides of the triangle. Since all sides of an equilateral triangle are equal, we can express the side length as: side = base.
Therefore, the area of each equilateral triangle becomes: area = (sqrt(3)/4) * (x^2 - x)^2.
3. Find the volume of each slice at a given value of x:
The volume of each slice is the area of the equilateral triangle multiplied by the dx (the infinitesimal width of the slice). So, we can express the volume of each slice as: dV = (sqrt(3)/4) * (x^2 - x)^2 * dx.
4. Integrate to find the total volume:
To find the total volume of the solid, we need to integrate the volume of each slice from x = 0 to x = 1. The integral becomes: V = ∫[0,1] (sqrt(3)/4) * (x^2 - x)^2 dx.
Simplifying the integral will give us the solution.
Now, let's solve the integral:
V = ∫[0,1] (sqrt(3)/4) * (x^2 - x)^2 dx
To find the integral, expand and simplify:
V = (sqrt(3)/4) * ∫[0,1] (x^4 - 2x^3 + x^2) dx
Integrate each term separately:
V = (sqrt(3)/4) * [(1/5)x^5 - (1/2)x^4 + (1/3)x^3] from x = 0 to x = 1
Substitute the limits:
V = (sqrt(3)/4) * [(1/5) - (1/2) + (1/3)] - (sqrt(3)/4) * [(0/5) - (0/2) + (0/3)]
Simplify further:
V = (sqrt(3)/4) * [(1/5) - (1/2) + (1/3)]
V = (sqrt(3)/4) * [(1 + 6 - 10)/30]
V = (sqrt(3)/4) * (-3/30)
V = -3(sqrt(3)/120)
V = -sqrt(3)/40
Therefore, the volume of the solid is -sqrt(3)/40 cubic units.