4. Let a, b e R, a < b. The following definite integral property is called...

Transcribed Image Text:

4. Let a, b e R, a < b. The following definite integral property is called "The union interval property". If f is integrable on [a, b], then for any c€ (a,b) [* f(x)dx = [*° f(x)dx + ["* f(x)dx. (a) Provide a sample diagram to illustrate this property. That is, care- fully draw and label an example picture to illustrate the union in- terval property. (b) (Challenge - not graded) Use only the Riemann definition of the definite integral to prove the Union Interval Property. (c) Prove or disprove: If f is integrable on [a, b] then for any c ER, ["* f(x)dx = [" f(x)dx + [" f(x)dx. a 4. Let a, b e R, a < b. The following definite integral property is called "The union interval property". If f is integrable on [a, b], then for any c€ (a,b) [* f(x)dx = [*° f(x)dx + ["* f(x)dx. (a) Provide a sample diagram to illustrate this property. That is, care- fully draw and label an example picture to illustrate the union in- terval property. (b) (Challenge - not graded) Use only the Riemann definition of the definite integral to prove the Union Interval Property. (c) Prove or disprove: If f is integrable on [a, b] then for any c ER, ["* f(x)dx = [" f(x)dx + [" f(x)dx. a