Understanding Integrands and Their Impact on Integrals: Explained

[tranj10]

I am having trouble understanding or visualizing how the integrand affects the plot of an integral.

So I know that

$\int_{0}^{10}\int_{0}^{10}\int_{0}^{10} 1 {dx}{dy}{dz}$

will give you the volume of a 10x10x10 cube. I am wondering what exactly you are getting when you integrate some thing like

$\int_{0}^{10}\int_{0}^{10}\int_{0}^{10} x^2 {dx}{dy}{dz}$

or even if the integrand was something like y^2 + z^2

If it is easier to understand what is happening with a single or double integral then I would be happy with an explanation from that.

 

[Mark44]

I am having trouble understanding or visualizing how the integrand affects the plot of an integral.

So I know that

$\int_{0}^{10}\int_{0}^{10}\int_{0}^{10} 1 {dx}{dy}{dz}$

will give you the volume of a 10x10x10 cube.

You might think of the integrand (1) as a function that gives the density at each point the 3D space. If we attach units to this function, say g/cm³, then the result is the mass of a 10 cm x 10 cm x 10 cm cube whose density is constant.

I am wondering what exactly you are getting when you integrate some thing like

$\int_{0}^{10}\int_{0}^{10}\int_{0}^{10} x^2 {dx}{dy}{dz}$

You could also think of the integrand here as being a density function define on R³, but the density varies only in the x-direction.

or even if the integrand was something like y^2 + z^2

If it is easier to understand what is happening with a single or double integral then I would be happy with an explanation from that.