Def 29.3 ∫𝑏𝑎𝑓(𝑥)𝑑𝑥=𝐼=lim𝑘∑𝜏𝑘𝑓=lim𝑅(𝑓,𝜉) if 𝑓 is integrable Prop 29.4 ∀𝜎 ∑𝜎𝑓≤∫𝑏𝑎𝑓(𝑥)𝑑𝑥≤∑𝜎𝑓 |∫𝑏𝑎𝑓(𝑥)𝑑𝑥|≤∑𝜎|𝑓| |∫𝑏𝑎𝑓(𝑥)𝑑𝑥|≤∫𝑏𝑎|𝑓(𝑥)|𝑑𝑥 Thm 30.3 for 𝑓, 𝑔 is integrable ∫𝑏𝑎(𝑓(𝑡)+𝜆𝑔(𝑡))𝑑𝑡=∫𝑏𝑎𝑓(𝑡)𝑑𝑡+𝜆∫𝑏𝑎𝑔(𝑡)𝑑𝑡 ∫𝑏𝑎𝑓(𝑡)𝑑𝑡=∫𝑐𝑎𝑓(𝑡)𝑑𝑡+∫𝑏𝑐𝑓(𝑡)𝑑𝑡 Thm 30.5 |∫𝑏𝑎𝑓𝑑𝑡|≤∫𝑏𝑎|𝑓|𝑑𝑡 proved by considering Φ(𝑦)=|𝑦| as a Lipschitz function. Def 31.5 By Oresme ∫10𝑥𝑚𝑑𝑥=1𝑚+1 Prop 41.3 ∫𝑏𝑎𝑓(𝑠)𝑑𝑠≤(𝑏−𝑎)sup𝑠𝑓(𝑠)