The base of a solid is the region bounded by y=2 √(sinx) and...

The base of a solid is the region bounded by $y=2 \sqrt{\sin x}$ and the $x$ -axis with $x \in[0, \pi / 2] .$ Find the volume of the solid given that cross sections perpendicular to the $x$ -axis are:
(a) equilateral triangles; (b) squares.

Key Concepts

Method of Cross Sections

This concept involves finding the volume of a solid by dividing it into thin slices (or cross sections) perpendicular to a given axis and then integrating the area of these slices over the range of the solid. The idea is to express the area of each cross section as a function of the variable along which the slices are taken and then sum these areas to get the total volume.

Definite Integration

Definite integration is used to sum an infinite number of infinitesimally thin slices to determine a total quantity, such as volume. In problems involving solids with known cross-sectional areas, the integral of the area function over the interval specified by the bounds of the solid yields the volume.

Geometric Formulas for Specific Cross Sections

When the cross sections of a solid have specific geometric shapes, such as equilateral triangles or squares, their area can be determined using known geometric formulas. For instance, the area of an equilateral triangle is given by (?3/4) multiplied by the square of its side length, while the area of a square is the side length squared.

Setting up the Limits of Integration and Variable Functions

Determining the appropriate limits of integration is crucial in volume problems. The region over which the volume is computed is often defined by the intersection of curves or boundaries. The function that describes the size or dimensions of the cross-sectional shape is expressed in terms of the integration variable, ensuring that the computed area accurately reflects the dimensions of the slice at any point along the solid.

Transcript

00:01 So hopefully you have an idea of what this graph looks like.

00:04 Because for sine of x, actually i think it just goes from 0 to pi over 2, because on the unit circle 0 to pi over 2 is up here.

00:16 So i could actually stop my graph sooner.

00:22 And we're looking at y equals 2 square root of sine of x.

00:29 And if you think about equilateral triangles, and i'll just go ahead and draw this in here and make it a little bit larger.

00:37 Actually, let me draw it this way, because a lot of people like to see the base on the ground, like on the flat part.

00:48 And then you can figure out what the height is, because you make a 30 -60 -90 triangle inside of here.

00:55 This distance is base divided by 2.

00:58 And inside you have a 30 -60 -90 triangle.

01:03 And that height is equal to the short leg times root 3.

01:06 So that base over root 2 is multiplied by root 3.

01:09 So the area of a triangle, which is normally just 1 half base times height, is now 1 half times the base.

01:18 And the height is now b over 2 root 3.

01:23 So this 2 times this 2 would give you 4.

01:27 So the volume for part a is going to be root 3 over 4 times the base squared.

01:35 And the bounds are still going to be from 0 to pi over 2, as was stated in the directions.

01:41 And then it's the function 2 square root of sine of x.

01:48 And so as i square each piece, which you write dx, i just go ahead and write 2 squared...

Please give Ace some feedback