the base of a solid is the region between the curve y=2 square root of sin x and the interval [0,pi] on the x-axis. the cross-sections perpendicular to the x-axis are equilateral triangles with bases running from the x-axis to the curve as shown in the accompanying figure.
Volume is nothing but the infinite sum of each individual area of infinitesimally small cross sections of the figure, which when stacked together perfectly describe the shape.
As given, the cross-sectional shape of the figure is an equilateral triangle. So, in order to find the volume, we must first find the areas of each individual equilateral triangle.
Notice that the base of the equilateral triangle is bounded by 2sqrt(sinx). Since it is described to be on the Euclidean plane, with the x-axis as the lower bound, the base of the triangle, at any point, has length 2sqrt(sinx).
You can derive the formula of an equilateral triangle, but I'll tell you: for a given base a, the area of an equilateral triangle is 4^-1*sqrt(3)a^2
As we know, the base of each equilateral triangle is described by the function. In other words, a = 2sqrt(sinx). We substitute, and arrive at this:
A(x)=4^-1*sqrt(3)(2sqrt(sinx))^2. Since in this case, we're working with injective functions and simple real numbers (which are integers) satisfy abelian groups: (ab)^2=a^2*b^2 (which is not always the case). Therefore, A(x)=4^-1*sqrt(3)*|4sin(x)|. We were only working with positive values anyway, so we can remove the absolute value. Now notice that the fours cause point-convergence to one. We get: A(x)=sqrt(3)sin(x).
Now, we integrate the area to find the volume. We integrate over the interval [0,pi]. So doing that, you get the integral [0,2pi] : sqrt(3)sinxdx. Remember, dx is the infinitesimally small width, delta in accordance with x, of each equilateral triangle cross section. Or, more accurately: in vector analysis, dx takes meaning as a differential form (roughly, something that behaves like an infinitesimally small piece of a curve).
We can take the constant 3^(1/2) outside of the integral. Now we are simply integration sin(x), which is a standard integral, equal to -cos(x)+C.
We don't care about the constant of integration in this case because we are working with a definite integral, and the constant of integration will be removed by a form of it of opposite sign, when summed.
So, now we use the fundamental theorem of calculus to solve the integral, and we arrive at: sqrt(3)(-cos(pi)+cos(0)) = sqrt(3)(1+1)=2sqrt(3)
That is the answer, 2sqrt(3).
To find the volume of the solid, we can use the method of cross-sectional area.
Step 1: Determine the equation for the area of an equilateral triangle.
The area of an equilateral triangle can be found using the formula A = (s^2 * √3) / 4, where s is the side length of the triangle.
Step 2: Express the side length of the equilateral triangle in terms of the variable x.
Since the base of the equilateral triangle runs from the x-axis to the curve y = 2√sin x, the length of the side of the equilateral triangle at any given x-value will be equal to the y-value of the curve at that x-value.
Therefore, the side length of the equilateral triangle, s, is equal to 2√sin x.
Step 3: Find the area of the equilateral triangle at each x-value.
Substitute the side length expression into the area formula: A = (s^2 * √3) / 4.
So, the area of the equilateral triangle at each x-value is A = (4 * (√sin x)^2 * √3) / 4 = (√3 * sin x).
Step 4: Integrate the areas of the cross-sections to find the volume.
To find the volume, we integrate the areas of the cross-sections over the interval [0, pi]:
V = ∫[0,pi] (√3 * sin x) dx
Now you can calculate the integral to find the volume of the solid.
To find the volume of the solid, we can use the method of cross-sections.
First, let's visualize the figure described in the problem. We have a curve y = 2√sin(x) and the interval [0, π] on the x-axis. The cross-sections of the solid are equilateral triangles, where the base of each triangle runs from the x-axis to the curve y = 2√sin(x), as shown in the figure.
To find the volume of the solid, we need to determine the area of each cross-section and then integrate it over the interval [0, π].
Let's start by finding the area of a single cross-section. The base of the triangle is the distance from the x-axis to the curve y = 2√sin(x), which is the y-coordinate at a given x-value.
Since the cross-section is an equilateral triangle, the height of the triangle is the same as the length of each side. To find this height, we can use the Pythagorean theorem.
The Pythagorean theorem states that for a right triangle with sides a, b, and hypotenuse c, the sum of the squares of the lengths of the two shorter sides is equal to the square of the length of the hypotenuse (a^2 + b^2 = c^2).
In our case, the height of the triangle is given by the y-coordinate of the point on the curve, and the length of the hypotenuse is the side of the equilateral triangle.
Let's denote the height of the triangle as h and the side of the equilateral triangle as s. We know that the side s is equal to the base of the triangle.
Using the Pythagorean theorem, we can write the following equation:
s^2 = h^2 + (1/2s)^2
Simplifying the equation, we get:
s^2 = h^2 + (1/4)s^2
Combining like terms, we get:
(3/4)s^2 = h^2
Taking the square root of both sides, we get:
s = 2√(3/4)h
Substituting the value of s into the equation, we have:
A = (1/2) base * height
= (1/2)(2√(3/4)h) * h
= √3/2 * h^2
Now that we have the area of a single cross-section, we need to integrate this over the interval [0, π] to find the total volume.
V = ∫[0,π] (√3/2 * h^2) dx
To express h in terms of x, we substitute y = 2√sin(x) into the equation:
h = 2√sin(x)
Therefore, the volume of the solid is:
V = ∫[0,π] (√3/2 * (2√sin(x))^2) dx
V = ∫[0,π] (2√3/2 * 4sin(x) dx)
V = 4√3 ∫[0,π] (sin(x) dx)
Using the integral property, we integrate sin(x) over the interval [0, π]:
V = 4√3 [-cos(x)] [0,π]
V = 4√3 ( (-cos(π)) - (-cos(0)) )
Since cos(π) = -1 and cos(0) = 1:
V = 4√3 (1 - (-1))
V = 4√3 * 2
V = 8√3 cubic units
Therefore, the volume of the solid is 8√3 cubic units.