Question

Volumes of Solids with Known Cross Sections

Answer 1

I am looking for the area of the square in terms of f(x) and g(x)
I know the area of a square is side squared
so the area is\[\left[ f(x)-g(x) \right]^2 \]

Answer 2

am I correct?
Also, I am not sure how to answer the second part about setting up the integral

Answer 3

\[\int\limits_{a}^{b}[f(x) - g(x)]^2dx\]

Answer 4

ok

Answer 5

\[\frac{ (f(x)-g(x))^2 }{ 2 }\]

Answer 6

The intersection of the two curves on the left seems to be on the y-axis and so a = 0.
b is the other intersection point of f(x) and g(x).

Answer 7

this is for the second one - isosceles right triangle

Answer 8

Yes.

Answer 9

can you help me do 2 others? I need a second to post the drawings

Answer 10

okay.

Answer 11

not sure about the equilateral triangle
but I know I is (1/2)base times height
(1/2)(f(x)-g(x)) not sure about the height

Answer 12

Since it is an equilateral triangle, all three sides are equal and all three angles of the triangle are 60 degrees. So the height is (f(x)-g(x)) * sin(60) = sqrt(3)/2 * (f(x)-g(x)).
Area of triangle = 1/2 * (f(x)-g(x)) * sqrt(3)/2 * (f(x)-g(x))
= sqrt(3)/4 * (f(x)-g(x))^2.

Answer 13

oh thanks, I was in the process of drawing the triangle since I was not sure how it was set up

Answer 14

You are welcome.

Answer 15

boy I would not even known to take this approach at all

Answer 16

no wonder I skip these problems. I am trying to get better at these types of problems

Answer 17

oh for the semicircle
I know to use (1/2) pi (radius^2)

Answer 18

radius is half of diameter
(1/2)(f(x)-g(x))

Answer 19

should be \[\frac{ 1 }{ 4 } \pi \left( f(x)-g(x) \right) ^2\]

Answer 20

@aum

Answer 21

Radius \(\Large r = \frac 12\)(diameter) = \(\Large \frac 12\{f(x)-g(x)\}\)
\(\Large r^2 = \frac14\{f(x)-g(x)\}^2\)
Area of semi-circle = \(\Large \frac 12\pi r^2\) = \(\Large \frac{ 1 }{ 8 } \pi \{ f(x)-g(x) \} ^2\)

Answer 22

I can't read this at all

Answer 23

only because site does not have equation editor active once a problem is closed