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.

Answer 1

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).