Integral of All Horrors

In response to the $2010$ Deep Horizon oil spill, ACME Inc. develops a machine that uses a high-tech particle beam to remotely seal oil leaks. To test the prototype, ACME has built a underwater test bed with $n$ independent oil leaks that are named L_1, L_2, \ldots, L_n.

Each leak L_i (1 \le i \le n) is characterized by parameters r_i and t_i. r_i is the initial oil leakage rate in litres per second ($L/s$). While L_i is being sealed by the particle beam, the leakage rate would linearly decrease, and reach $0$ $L/s$ (i.e. become sealed) after t_i seconds of continual operation.

For example, let L_1 be characterized by r_1 = 20 and t_1 = 4. Initially oil spew at a rate of $20$ $L/s$. After $1$ second of sealing, the rate reduces to $15$ $L/s$; after $2$ seconds, the rate is $10$ $L/s$; after $3$ $s$, we have $5$ $L/s$; and after $4$ $s$ the leak is sealed. Throughout these $4$ seconds, the total volume of oil that leak from L_1 is $20$ ($L/s$) $\times$ $4$ ($s$) $/$ 2 = 40 $L$ (area of a triangle). Meanwhile, other leaks continue to spew oil, and are unaffected until the particle beam is directed at them.

You are responsible for demonstrating the machine. At t = 0, you activate the beam to seal leaks. The particle beam takes negligible time to target a leak, and is never idle. Therefore every leak is sealed at t = t_1 + t_2 + \ldots + t_n (and then everyone applauds). However, a lot of oil would be leaked during the test. The precise amount depends how you choose to target the particle beam throughout the demonstration.

Given parameters for L_1, L_2, \ldots, L_n, your task is to minimize total volume (in litres) of oil leakage between t = 0 (particle turned on) and t = t_1 + t_2 + \ldots + t_n (all leaks sealed). Note that there are multiple test cases (all independent from one another).