Continuity and Integral Problem

[Jason76]

Which of the following integrals have the same value? Considering \(\displaystyle f\) is continuous for all of \(\displaystyle x\)


a. \(\displaystyle \int^{b}_{a} f(x) dx\)

b. \(\displaystyle \int^{b-a}_{0} f(x + a) dx\)

c. \(\displaystyle \int^{b + c}_{a + c} f(x + c) dx\)

Answer: a and b - Why? Hint

 

[Deleted member 4993]

Which of the following integrals have the same value? Considering \(\displaystyle f\) is continuous for all of \(\displaystyle x\)


a. \(\displaystyle \int^{b}_{a} f(x) dx\)

b. \(\displaystyle \int^{b-a}_{0} f(x + a) dx\)

c. \(\displaystyle \int^{b + c}_{a + c} f(x + c) dx\)

Answer: a and b - Why? Hint

Assume:

F'(x) = f(x)

\(\displaystyle \displaystyle \int f(x) dx = F(x) + C\)

\(\displaystyle \displaystyle \int_b^a f(x) dx = F(a) - F(b)\)

now continue.....

 

[HallsofIvy]

Hint: substitution. In (b) let u= x+ a. In (c) let u= x+ c