Find the volume of the region

[BlBl]

This is probably dead simple and I'm just not seeing it.

Under the graph of \(\displaystyle f(x,y) = x+y\) and above the region \(\displaystyle y^2 \leq x\), \(\displaystyle 0 \leq x \leq 9.\)​

Here's the integration I've set up so far:

\(\displaystyle \int\limits_{0}^{9}\int\limits_{0}^{3} (x+y) \ dy\,dx \)

I got 162 from this, but that's not the answer I'm looking for (it should be 117.45), so that means that I set the integral up incorrectly. But what should I have done? Usually when I have these constraints on an integration problem I have a mention or implication of z, but here there doesn't seem to be anything.

Help?

 

[Deleted member 4993]

This is probably dead simple and I'm just not seeing it.

Under the graph of \(\displaystyle f(x,y) = x+y\) and above the region \(\displaystyle y^2 \leq x\), \(\displaystyle 0 \leq x \leq 9.\)​

Here's the integration I've set up so far:

\(\displaystyle \int\limits_{0}^{9}\int\limits_{0}^{3} (x+y) \ dy\,dx \)

I got 162 from this, but that's not the answer I'm looking for (it should be 117.45), so that means that I set the integral up incorrectly. But what should I have done? Usually when I have these constraints on an integration problem I have a mention or implication of z, but here there doesn't seem to be anything.

Help?

The limits on y should be -√x to +√x then limit on x is 0 to 9.

 

[BlBl]

The limits on y should be -√x to +√x then limit on x is 0 to 9.

Thanks for the reply Subhotosh. I still don't get the answer that the book is giving me for this question with that. I end up with 194.4 or 972/5.

Maybe the book is wrong? Maybe my integration technique is wrong? I don't know at this point.

Did you get 117.45 when you put in -√x to +√x?

 

[thepillow]

Thanks for the reply Subhotosh. I still don't get the answer that the book is giving me for this question with that. I end up with 194.4 or 972/5.

Maybe the book is wrong? Maybe my integration technique is wrong? I don't know at this point.

Did you get 117.45 when you put in -√x to +√x?

Yes, you're right about the result of the double integral:

\[ \int_0^9 \int_{-\sqrt{x}}^{\sqrt{x}} x+y \,dy\,dx = \frac{972}{5} = 194.4\]

I think that your book's answer of 117.45 comes from this integral:

\[ \int_0^9 \int_{0}^{\sqrt{x}} x+y \,dy\,dx = \frac{2349}{20} = 117.45 \]

The only difference is that y is starting at 0. Maybe they meant to give the region as 0≤y²≤x.

Hope that helps.

 

[BlBl]

Yes, you're right about the result of the double integral:

\[ \int_0^9 \int_{-\sqrt{x}}^{\sqrt{x}} x+y \,dy\,dx = \frac{972}{5} = 194.4\]

I think that your book's answer of 117.45 comes from this integral:

\[ \int_0^9 \int_{0}^{\sqrt{x}} x+y \,dy\,dx = \frac{2349}{20} = 117.45 \]

The only difference is that y is starting at 0. Maybe they meant to give the region as 0≤y²≤x.

Hope that helps.

Thanks thepillow, I came to this conclusion myself yesterday after I revisited the problem. The professor has apparently said that they need to redo this question.