Finding the bounds of a multivariable integration to compute the volume between two functions?

Question

I have two multivariable functions and I want to find the bounds to integrate.
$f(x,y)=x^2+4y^6$
$g(x,y)=\sin{x}+\frac{1}{5}y+4$
How can I determine the bounds to integrate. Thereby, I want to integrate the volume of $f(x,y)$ under $g(x,y)$ which limits my integration on the $z$ axis. But how can I determine the $x$ and $y$ bounds? Is it possible to determine where those two functions intersect or is that approach wrong? Maybe you can get a picture of what I am talking about if you take a look at the image. Help would be much appreciated. enter image description here

Answer 1

Precision on the problem

If your problem is well understood you seek for a given $(x,y)\in \mathbb{R}^2$. Note $z=g(x,y)$

$$\int_0^{z=\sin(x)+\frac{1}{5}y+4}f(x,y)dxdy \ \text{is not appropriate}$$

because $g(x,y)$ is fixed as well as $(x,y)$. But $dx$ and $dy$ are moving

Here you choose to make $z$ a function of $(x,y)$.

First

Actually wanting to write it properly to calculate it properly.

$$ \int_0^{z=\sin(x)+\frac{1}{5}y+4}f(u,v)dudvdw$$

Secondly

You have to fix a bound for $x$ and $y$ let's say respectively $a$ and $b$ :

$$ \int_a^x \int_b^y\int_0^{z=\sin(x)+\frac{1}{5}y+4}f(u,v)dudvdw$$

Conclusion

It is very important to write the formula of what you want so you will be able to easily determine what is need to be defined.

Hope it helped.