Question

4. Let a, b ? 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) $\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$. (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 ? R, $\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$.

          4. Let a, b ? 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)
$\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$.
(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 ? R,
$\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$.

4. Let a, b ? 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)
^b f(x)dx = ^c f(x)dx + ^b 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 ? R,
^b f(x)dx = ^c f(x)dx + ^b f(x)dx.

Added by Steve S.

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