#todo > [!NOTE] Lemma > Let $f:[a,b]\to \mathbb{R}$ be [[Riemann integration|integrable]]. Then $\int_{a}^{b} f(x) \, dx = \int_{a}^{b} \int_{0}^{f(x)} dy \, dx $ **Proof**: Let $\{ x_{0},x_{1},\dots,x_{n} \}$ be a [[Finite Partition of Closed Real Interval|finite partition]] of $[a,b]$ into $n$ equal intervals. By ..., $\int_{a}^{b} \, dx = \lim_{ n \to \infty } \sum_{i=0}^{n-1} f(x_{i}) (x_{i+1} - x_{i}) = \lim_{ n \to \infty } \lim_{ m \to \infty } \sum_{i=0}^{n-1} \sum_{j=0}^{m_{i}-1} (y_{j+1}-y_{j}) (x_{i+1}-x_{i}) = \int_{a}^{b} \int_{0}^{f(x)} \, dy \, dx $ **Proof**: By [[Fundamental theorem of calculus]], $\int_{a}^{b} \int_{0}^{f(x)} \, dy \, dx = \int_{a}^{b} \big [y\big]_{0}^{f(x)} \, dx =\int_{a}^{b} f(x) \, dx .$