Use a double integral to find the volume of the indicated solid

[iRaid]

Homework Statement

Use a double integral to find the volume of the indicated solid.

Homework Equations

The Attempt at a Solution

I can't find what I did wrong, it seems like a simple problem...
$$\int_0^2 \int_0^x (4-y^{2})dydx=\int_0^2 4x-\frac{x^{3}}{3}dx$$
$$=2x^2-\frac{x^{4}}{12}|_0^2=8-\frac{16}{12}=\frac{20}{3}$$

 

[STEMucator]

Homework Statement

Use a double integral to find the volume of the indicated solid.

Homework Equations

The Attempt at a Solution

I can't find what I did wrong, it seems like a simple problem...
$$\int_0^2 \int_0^x (4-y^{2})dydx=\int_0^2 4x-\frac{x^{3}}{3}dx$$
$$=2x^2-\frac{x^{4}}{12}|_0^2=8-\frac{16}{12}=\frac{20}{3}$$

One of your limits for ##y## is wrong.

 

[iRaid]

One of your limits for ##y## is wrong.

I'm not seeing it, sorry.

 

[STEMucator]

If you graph the region in the x-y plane, it should look something like this:

http://gyazo.com/aedf21fdd2006d58eabea7d5b3324065

Suppose you hold ##x## fixed and allow ##y## to vary. Then clearly from the above graph ##0 ≤ x ≤ 2## and ##x ≤ y ≤ 2##.

Try letting ##y## be fixed and allowing ##x## to vary now. Do you get the same result?

 

[iRaid]

If you graph the region in the x-y plane, it should look something like this:

http://gyazo.com/aedf21fdd2006d58eabea7d5b3324065

Suppose you hold ##x## fixed and allow ##y## to vary. Then clearly from the above graph ##0 ≤ x ≤ 2## and ##x ≤ y ≤ 2##.

Try letting ##y## be fixed and allowing ##x## to vary now. Do you get the same result?

Ah right, I feel dumb now.

Thank you.