MAT137Y5 Chapter Notes - Chapter integrabitiy: Integral Symbol, Binomial Theorem, Triangle Inequality

• School: UTM
• Department: Mathematics
• Course: MAT137Y5
• Professor: Ali Mousavidehshikh

Document Summary

Everything you need to know about integration you already know the intuition behind the logic of integrals. Basi(cid:272)all(cid:455), it(cid:859)s a (cid:271)u(cid:374)(cid:272)h of (cid:396)e(cid:272)ta(cid:374)gles that approximate the area under a curve, and as the number of rectangles approaches infinity, this approximation approaches the area under the curve. However, proving integrability puts you in the same spot as you were with proving differentiability1. The link between derivatives and integrals has not (cid:271)ee(cid:374) (cid:373)ade (cid:455)et, a(cid:374)d (cid:455)ou(cid:859)ll see (cid:449)h(cid:455) (cid:449)he(cid:374) (cid:455)ou p(cid:396)o(cid:448)e a fu(cid:374)(cid:272)tio(cid:374) is i(cid:374)teg(cid:396)a(cid:271)le it(cid:859)s (cid:374)ot intuitive when doing the proof of integrability that it could link to derivatives. Mathematically, proving integrability in mat137 in 2017-18 uses darboux sums, which are more elegant version of riemann sums. I(cid:374)(cid:272)e (cid:455)ou al(cid:396)ead(cid:455) ha(cid:448)e a(cid:374) i(cid:374)tuitio(cid:374), i(cid:859)ll ju(cid:373)p i(cid:374)to the defi(cid:374)itio(cid:374) a(cid:374)d so(cid:396)t thi(cid:374)gs out from there. In mat137, you need to know how to prove integrability. This means you need to translate your intuitions into math.