Volume: The Disk/Washer Method

[layd33foxx]

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. (Round your answer to three decimal places.) y = e^−x
y = 0
x = 0
x = 7

So here's what I've done:I have the wrong answer idk why

R(x)=e^-x
V= ∫ ⁷ (e^-x)²dx
=" " (e^-2x)dx
=[(π/2) e^-2x]
=(π/2)(7- e^-2x)
=10.783

 

[tkhunny]

Sloppy, sloppy. Small wonder you wandered off a bit.

When you created "V", where did \(\displaystyle \pi\) go? It shows up later.

When you created "V", was there a lower bound?

When you found the anti-derivative, where did the "-" go? That's "-2" in that exponent.

When you began to substitute into your antiderivative, why is 'x' still in there? Is this a side effect of the missing lower bound?

\(\displaystyle \int_{0}^{7}\pi (e^{-x})^{2}\;dx = \frac{\pi}{2}(1-e^{-14}) = 1.5707950206\) -- ALMOST \(\displaystyle \frac{\pi}{2}\)

It is almost always a good idea to do it the other way, just to make sure you are getting it.

\(\displaystyle \int_{e^{-7}}^{1}\pi\cdot y\cdot (-ln(y))\;dy+\pi\cdot (e^{-7})^{2}\cdot 7\)

Please follow both of these through until you see their equivalence.

 

[layd33foxx]

i have to review my antiderivative I think thats why I'm lost:sad:

 

[burakaltr]

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. (Round your answer to three decimal places.) y = e^−x
y = 0
x = 0
x = 7

So here's what I've done:I have the wrong answer idk why

R(x)=e^-x
V= ∫ ⁷ (e^-x)²dx
=" " (e^-2x)dx
=[(π/2) e^-2x]
=(π/2)(7- e^-2x)
=10.783

Volume of a single cylinder Pi*y**2*dx

so

PI*integral ( y**2) dx|from x=0 to x=7

PI * int(e**-2x)dx

integral is : e**(-2x)
----------------------------------------
-2

e**-14 - 1
-----------------
-2

is almost 0.5

which leads to conclusion that the VOLUME is


approx.


0.5 * PI

 

[tkhunny]

i have to review my antiderivative I think thats why I'm lost:sad:

That is correct, but please pay more heed to the first warning, "Sloppy, sloppy. Small wonder you wandered off a bit.
"

 

[renegade05]

Volume of a single cylinder Pi*y**2*dx

so

PI*integral ( y**2) dx|from x=0 to x=7

PI * int(e**-2x)dx

integral is : e**(-2x)
----------------------------------------
-2

e**-14 - 1
-----------------
-2

is almost 0.5

which leads to conclusion that the VOLUME is


approx.


0.5 * PI

What does ** mean?

If I didn't know what I was doing, your solution would honestly have confused me more than helped.

You should use Latex for solution, it's clear and has great formatting.

Just my opinion.

 

[burakaltr]

What does ** mean?

If I didn't know what I was doing, your solution would honestly have confused me more than helped.

You should use Latex for solution, it's clear and has great formatting.

Just my opinion.

Yes OK. It is FORTRAN notation..

 

[tkhunny]

Being an old FORTRANer, I didn't even recod the notation as unusual.