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 bounded by the graphs of $y=3 x, y=6,$ and $x=0 .$ The cross-scctions perpendicular to the $x$ -axis are
a. rectangles of height 10 .
b. rectangles of perimeter 20 .

Key Concepts

Slicing Method

The slicing method involves breaking down a solid into a series of thin, parallel slices perpendicular to an axis. The volume of the solid is obtained by summing the volumes of these slices, which are approximated as having uniform cross-sectional area, across the entire interval. This method transforms a three?dimensional problem into a one?dimensional integration problem.

Integration of Cross-sectional Areas

After determining the area of a typical cross section as a function of position along the axis, integration is used to add up these areas over the entire range of the solid. This integral represents the total volume, with the cross-sectional area serving as the integrand and the limits defined by the projection of the base onto the integration axis.

Determining Integration Limits

The limits of integration are established by projecting the base region of the solid onto the axis along which the slices are taken. This involves finding the boundaries of the base region in that one-dimensional projection, and these boundaries become the limits for the integral that calculates the volume.

Geometric Analysis of Cross Sections

The problem specifies cross sections with particular geometric properties, such as those defined by a fixed height, fixed perimeter, or other constraints. Analyzing these properties allows one to express the area of each cross section in terms of the position variable, which is essential for setting up the integral that calculates the volume of the solid.

Transcript

00:01 In part a, we'll be taking these blue cross sections perpendicular to the x -axis, which are rectangles of height 10.

00:10 And so we'll kind of put post -it notes where each of those blue lines are that pop out of the page towards ourselves.

00:17 And so we're going to have the area is going to be base times height, or we could say base, which is the, are those lines in blue times the height, which is going to be 10, according to point.

00:31 Part a.

00:33 Okay.

00:34 And now we're going to add up each of those from left to right.

00:38 And so the volume is going to be from zero to two.

00:44 The expression for the base is it's going to be the top function minus the bottom function, which is 3x.

00:58 And that's going to in turn give us that blue distance here.

01:03 So 6 minus 3x would give us each of these blue segments there, or the base.

01:11 And then we're going to multiply each of those by 10.

01:14 And that'll add up all of the rectangles from left to right, from 0 to 2.

01:21 So simplifying this, we'll get the integral from 0 to 2 of 60, distributing the 10 minus 30x.

01:34 And then taking the anti -derivative, we'll end up with 60x minus 30 x squared divided by 2, which would simplify it at 15.

01:54 And from here, we could go ahead and plug in two first and then zero, but zero would just cancel everything.

02:02 So our final answer is going to be 60 minus 4 times, oh, let's see, it's going to be 120 minus 60.

02:15 That's it.

02:16 And that'll end up giving us a volume of 60 for part a.

02:23 And then moving on to part b...

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