Understand Clearly — Single and Double Integration

First, a simple example — one-variable integral:

Mathematical notation:

\( \int x^2 \, dx \)

In simple words: if you want to find the area/antiderivative of the function \(x^2\), then

\[ \int x^2 \, dx = \frac{x^3}{3} + C \]

Two variables — Double Integration

When a function depends on two variables — like \(x\) and \(y\) — we take the double integral. Its general form is:

\[ \iint_R f(x,y)\,dx\,dy \]

Here, \(R\) is the region over which you are integrating. Often, we write it as iterated integrals — first integrate with respect to one variable, then the other. For example:

\[ \iint_R f(x,y)\,dx\,dy = \int_{y=a}^{b}\left(\int_{x=g_1(y)}^{g_2(y)} f(x,y)\,dx\right) dy \]

Or in the other order:

\[ \iint_R f(x,y)\,dx\,dy = \int_{x=c}^{d}\left(\int_{y=h_1(x)}^{h_2(x)} f(x,y)\,dy\right) dx \]

Note: Switching the integration order may change the limits — so make sure you understand the region \(R\) first.

Step-by-Step Example — Double Integration (One Example)

Function:

\( f(x,y)=x^{2}+3x y^{2}+x y \)

Integrate with respect to \(y\) (treat \(x\) as constant):

1. \( \displaystyle \int x^{2}\,dy = x^{2}y \)
2. \( \displaystyle \int 3x y^{2}\,dy = 3x\cdot\frac{y^{3}}{3} = x y^{3} \)
3. \( \displaystyle \int x y\,dy = x\cdot\frac{y^{2}}{2} = \frac{x y^{2}}{2} \)

Final result:

\[ \int (x^{2}+3xy^{2}+xy)\,dy = x^{2}y + x y^{3} + \frac{x y^{2}}{2} + C \]

Note: If this is part of a double integral, the next step is to integrate the above expression with respect to \(x\) (or evaluate according to the given x-limits).