Be $\Sigma \subset \mathbb{CP}^2$ the zero loci of $f(x,y)= y^2 - x(x-1)\prod_i (x-\lambda_i)$, where $(x,y)$ are affine coordinates.
$\Sigma$ is a smooth algebraic curve, and also a Riemann surface, of genus 2. There are two holomorphic differentials, given by $\omega_i \frac{dx}{y} x^{i-1}, \quad i=1,2$. Topologically, $\Sigma$ is a double torus, with a singular homological basis given by $H_1(\Sigma) = <\alpha_1,\alpha_2,\beta_1,\beta_2>_{\mathbb{Z}}$, with intersection numbers $\alpha_i \wedge \beta_j = \delta_{ij}$ and zero otherwise.
We can visualize $\Sigma$ as a double branched cover of $\mathbb{CP}^1$, with the covering map $(x,y(x)) \to x$, with branching points at $0,1,\infty,\lambda_i$. We have branching cuts joining the pair of points $[0,1],[\lambda_1,\lambda_2],[\lambda_3,\infty]$. (This choice is arbitrary, depending on how we define the square root of $x (x-1) \prod_i (x-\lambda_i)$).
We can visualize $\alpha_i$ as a loop over one copy of $\mathbb{CP}^1$ circling one of these cuts. Let us say that $\alpha_1$ circles $[0,1]$ and $\alpha_2 $ circles $[\lambda_1,\lambda_2]$.
How can we compute $\int_{\alpha_i} \omega_j$?
Alternatively, we have two holomorphic differentials $\tilde{\beta}_j$ (Poincarè's duals to the cycles $\beta$) so that for them $\int_{\alpha_i} \tilde{\beta}_j = \alpha_i \wedge \beta_j = \delta_{ij}$, and which are a basis for holomorphic differentials. How can we write $\omega$ in terms of $\tilde{\beta}$?