Question

volume

Answer 1

of which shape????

Answer 2

I like volume of pie!

Answer 3

Mmm \(\Large\pi ^3\)!

Answer 4

#7

Answer 5

sooo

Answer 6

first you must find the 0s of the function which in this case is just x = 2 and x = -2

these are the bounds of integration.

so \[\int\limits_{-2}^{2} (-x^2 +4)dx \]

thats the area of the base, but for each crossection you are using f(x) to get the volume so in the case of the square it would be

\[\int\limits_{-2}^{2} (-x^2 +4)^2dx \] because each iteration of the integral would give you a slice, and the integral adds it all together. You can use the same principle for the other crossections.

Answer 7

mhm

Answer 8

so maybe you can see in my poorly drawn image how the one side of the square is determined by f(x) and you would have a bunch of squares stacked next to eachother which as the change in x approaches 0 which is why you have to integrate. does that make sense?

Answer 9

yes i see it

Answer 10

then what just solve teh integral u have above?

Answer 11

yes, thats the integral for the square cross-sections.

Answer 12

how come it mentions 3 different shapes tho

Answer 13

512/15?

Answer 14

well, for the semi-circle cross-sections you would have to do a different integral. since the area of a circle is \[\pi r^2\] and the radius is f(x)/2 you then have the integral:\[\int\limits_{a}^{b}\pi(f(x)/2)^2dx\]

Answer 15

so thats just times pi and divided by 2?

Answer 16

yeah, you have to realize that there is no set way of doing these, it changes with every single shape. The integral is something you have to derive on the spot from the area of the cross-section.

Answer 17

lol oh ok whoops

Answer 18

ok i got 8pi over 3

Answer 19

what about rectangle with height half its base

Answer 20

well think about it, if f(x) is the base, then the height is f(x)/2

Answer 21

so the area of the rectangle is f(x) * f(x)/2

Answer 22

ooooooooo same bounds tho right

Answer 23

yes

Answer 24

no pi cuz its arectangle correct

Answer 25

n its not squared?

Answer 26

mhm

Answer 27

ok an addition question

Answer 28

well it would be squared, f(x)*f(x) = f(x)^2 so it would be (f(x)^2)/2

Answer 29

choose one of the cross sectionsf rom this problem and draw them perpendcular to the y axis. will that change the volume?

Answer 30

oh ok

Answer 31

instead of using f(x) you would use the inverse f^(-1)(y) which in this case would be sqrt(-y + 4) you would then do exactly what you just did and see if you get a different answer.

Answer 32

i got a different answer... was i supposed to?

Answer 33

which one did you try?

Answer 34

wait, yes , I think you were supposed to get a different answer no matter which one you chose.