3) Let ( f(x)=e^2 x ). Let ( R ) be the region in the first...

3) Let ( f(x)=e^{2 x} ). Let ( R ) be the region in the first quadrant bounded by the graph of ( f(x) ) and the vertical line ( x=1 ), as shown in the figure below. a. Set up the integral (do NOT evaluate) used to find the volume if ( R ) is rotated around the ( x )-axis. b. Region ( mathrm{R} ) forms the base of a solid with equilateral triangle cross sections that are perpendicular to the ( x )-axis. Set up the integral but do not evaluate, that is used to find the volume of the solid. c. Set up the integral, but do not evaluate, that is used to find the volume if region ( R ) is rotated around the line ( y=8 )

Transcript

00:01 When you're revolving around an axis like we have on this problem, so i like that they even give you the ordered pair 1e to the second power.

00:10 Okay, so we're revolving around the x -axis first, so what we're doing is we're creating these disks.

00:16 And i don't know if you can tell, but this is a disk, and it's the area of a circle, which is just pi r squared, but there's an infinite number of circles in here.

00:27 So that's why your first answer for letter a you have a pi in front let me write v equals volume is equal to the pi just goes in front the integral is from those x values going from zero to one and then the function we're looking at is e to the 2x being squared in terms of x and they tell you not to solve it so we'll just leave it like that okay so then the next one is what if we have an equilateral triangle cross sections perpendicular to the x -axis? so what that, i just haven't memorized, maybe i should explain why, that it's root 3 over 4 and then the integral from a to b of the function being squared dx.

01:19 So if you want an explanation why it's root 3 over 4 is if you have an equilateral triangle, that's a terrible triangle, but the base is this length.

01:29 Well the area of a triangle is one half the base times the height.

01:34 Well how is the height related to this problem? well we have a 30 -60 -90 triangle in here.

01:40 See right there is my height and this piece is half the base and you get from the short leg to the long leg of the height is you take that piece one half the base times it by root 3 and so if i substitute that in for h, half and and half gives me four and that root three and b times b is the base squared so i don't know if you needed to see all that but that's going to be your answer root three over four the integral going from zero to one and again it's our function e to the 2x being squared dx and for the second time they don't ask you to solve that so let's stop there and then set up an integral but do do not evaluate.

02:28 If we are revolving about the line, y equals 8.

02:31 Okay, so i can go up.

02:33 Y equals 8 is barely above e squared...

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