I want to calculate the volume of the unit sphere: $\{(x,y) \in\mathbb{R} \,|\, x^2+y^2 \leq 1\} = K$
I am supposed to use iterative integration. We have defined the volume as:
$\int_\mathbb{R^2}\,\, 1_K \,\,d\lambda^2$
I can write this as an iterative integral:
$\int_a^b \int_c^d \,\,1 \,\,dx_1 dx_2$, where a,b and c,d are the bounds for $K$ so that $(a,c) \leq (x,y) \leq (b,d) \Leftrightarrow (x,y) \in K$.
My approach was, for symmetrical reasons, to first write it as $4 \int_0^b \int_0^d \,\,1 \,\,dx_1 dx_2$ and then choose $d = \sqrt{1-b^2}$ but I just can't get to the result that would be correct: $\pi$
I tried to search for similar questions and they all used some spherical transformation without explaining what they do exactly. We didn't have spherical transformation yet. We had transformation though, but I don't see how they get these transformation equations and why they can be used.
My approach: $4 \int_0^b \int_0^{\sqrt{1-b^2}} \,\,1 \,\,dx_1 dx_2 = 4 \int_0^b \sqrt{1-b^2} \,\,dx_2 = 4b\sqrt{1-b^2}$
But now I'm not sure how to continue, how to choose $b$. My guess would be $b=1$ but then the result would be $0$.