Find the volumes of the solids in Exercises 1-10. The base...

Find the volumes of the solids in Exercises $1-10$. The base of a solid is the region between the curve $y=2 \sqrt{\sin x}$ and the interval $[0, \pi]$ on the $x$ -axis. The cross-sections perpendicular to the $x$ -axis are a. equilateral triangles with bases running from the $x$ -axis to the curve as shown in the accompanying figure. b. squares with bases running from the $x$ -axis to the curve.

Find the volumes of the solids in Exercises $1-10$.
The base of a solid is the region between the curve $y=2 \sqrt{\sin x}$ and the interval $[0, \pi]$ on the $x$ -axis. The cross-sections perpendicular to the $x$ -axis are
a. equilateral triangles with bases running from the $x$ -axis to the curve as shown in the accompanying figure.
b. squares with bases running from the $x$ -axis to the curve.

Key Concepts

Area Formulas for Geometric Shapes

When the cross sections of a solid are standard geometric shapes, such as equilateral triangles or squares, known area formulas for these shapes are used. For an equilateral triangle, the area is expressed in terms of its side length using the formula (sqrt(3)/4) times the square of the side, and for a square, it is simply the square of the length of its side. These formulas link the dimensions determined by the base region to the computed area of each slice.

Volume by Cross-sectional Integration

This concept involves finding the volume of a solid by integrating the area of its cross sections along an axis. By slicing the object perpendicular to a given axis, the total volume can be computed as the integral of these individual cross-sectional areas over the interval corresponding to the object's extent. This method is particularly useful when the cross-sectional shape varies along the axis.

Defining the Base Region Using Functions

The base of the solid is defined by a function that describes the boundary of the region in the plane. In problems involving volumes by cross-sectional integration, this function determines the size or dimensions of the cross-sectional shape at each point along the integration axis. Understanding this relationship is essential to correctly setting up the integral that will yield the volume of the solid.

Transcript

00:01 Oh, hello, numid.

00:02 Welcome back.

00:03 All right, so today, what are we doing? we're doing, come on, penn.

00:10 6 .1, chapter 6, section 1, number 5.

00:16 All righty, and says the base of the solid of the region, find a, okay, base, the base of a solid is the region between the curve, y equals 2 rad sine x, and the interval from 0 to pi inclusive on the x -axis.

00:32 The cross sections perpendicular to the axis are part a, okay, part a.

00:41 Equilateral triangles with bases running from the x -axis to the curve as shown in the figure above.

00:46 Okay.

00:47 So the base of this equilateral triangle has a length of y equals 2 red sinex.

00:55 What is the area of the face of that triangle? because what we want to integrate is zero the face times, you know, a little thickness, dx, for each little triangle, which has a different base, as we span y equals 2, rad sinex from zero to pi.

01:12 So if you think about it, let's see, can i draw triangles with this thing? no, not that.

01:18 No, no, no, cancel.

01:19 Where's the thing for triangles? here it is.

01:21 Here's a triangle.

01:23 Okay.

01:24 All right, well, there's my triangle.

01:27 Can i make it bigger? there you come.

01:29 All right, so let's get back to the pens.

01:32 Pen, give me a pen.

01:33 No triangles.

01:35 All right, so if this is why the base, is y which equals 2 red sine x and that's the face of the triangle and it's coming up perpendicular from the x axis from the x x x up to y equals sign red uh 2 sine red x what's the area of this triangle while the area of triangle is based on height so there's the base 2 red sine x what's the height what is h so this is the base the base is this what's the height well if you think about it i wish this wasn't filled in could i draw on top of this color.

02:15 Let's see.

02:16 Here's the height, right? okay, there you go.

02:20 So think about it.

02:21 And equilateral triangle is pretend this is equilateral.

02:24 Actually, it doesn't look like an equilateral triangle.

02:25 It does.

02:25 It looks like that sassis.

02:26 But equilateral triangles are all five sassi's.

02:28 If you draw the altitude of an is sassi's triangle, not only do you get a median to the base, so this piece here is half of b.

02:37 This piece here is half of y...