MAT136H1 Lecture Notes - Lecture 4: Riemann Sum

• School: UTSG
• Department: Mathematics
• Course: MAT136H1
• Professor: Paul Milgram

Document Summary

Mat136 lecture 4 fundamental theorem of calculus (i) Last time: we introduced the definite integral of a function (cid:2188):[(cid:2183),(cid:2184)] . =((cid:2187)(cid:2183) (cid:2196) (cid:2187) (cid:2196) (cid:2183)(cid:3409)(cid:2206)(cid:3409)(cid:2184) )=(cid:2194)(cid:2195)(cid:2196) (cid:4684) (cid:2184) (cid:2183)(cid:2196) (cid:2207)=(cid:2188)(cid:4666)(cid:2206)(cid:4667) (cid:2193)=(cid:2778) (cid:2777) More generally, (for positive and negative functions): red = region a1, green = region a2, blue = region a3, the area under the curve can be given by: (cid:3029) =, (cid:2870)+(cid:2871: a1 and a3 have positive area, and a2 have negative area. If a region lives above the x-axis, its area is positive, if it lies below the x axis, its area is negative is the net signed area (cid:271)e(cid:272)ause (cid:449)e"(cid:396)e taki(cid:374)g the net sum of signs (cid:3029)(cid:3028) We could do it with riemann sums, but it would be really hard. =(cid:2869: this is scary a(cid:374)d (cid:448)e(cid:396)(cid:455) diffi(cid:272)ult so it"s easie(cid:396) to see the (cid:395)uestio(cid:374) as geo(cid:373)et(cid:396)(cid:455) Properties of definite integrals: (cid:1858)(cid:4666)(cid:4667) (cid:3028)(cid:3029)