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RJLiberator

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True or False? Let a and b be real numbers, with a < b, and f a continuous function on the interval [a, b].

a) If a=b then [itex]\int^{b}_{a} f(x)dx = 0[/itex]

My answer: This is TRUE, because while this integral would have a height, it would NOT have a width and area being l*w will result in 0.

b) If [itex]a \neq b[/itex], then [itex]\int^{b}_{a}f(x)dx \neq 0[/itex]

My answer: This is FALSE, because there will exist a height and a width, however, half of the area can be negative area and half of the area can be positive area canceling each other out and creating 0 area

c) If [itex]a \neq b[/itex], then [itex]\int^{b}_{a}f(x)dx = 0, then f(x) = 0 for all x \in [a,b] [/itex]

My Answer: This is FALSE, however, I am having trouble finding an example.

D) If [itex]a \neq b[/itex], then [itex]\int^{b}_{a}|f(x)|dx = 0, then f(x) = 0 for all x \in [a,b] [/itex]

My Answer: EDIT: This is True.

Correct answers?

Thanks for checking in with me and guiding me.

a) If a=b then [itex]\int^{b}_{a} f(x)dx = 0[/itex]

My answer: This is TRUE, because while this integral would have a height, it would NOT have a width and area being l*w will result in 0.

b) If [itex]a \neq b[/itex], then [itex]\int^{b}_{a}f(x)dx \neq 0[/itex]

My answer: This is FALSE, because there will exist a height and a width, however, half of the area can be negative area and half of the area can be positive area canceling each other out and creating 0 area

c) If [itex]a \neq b[/itex], then [itex]\int^{b}_{a}f(x)dx = 0, then f(x) = 0 for all x \in [a,b] [/itex]

My Answer: This is FALSE, however, I am having trouble finding an example.

D) If [itex]a \neq b[/itex], then [itex]\int^{b}_{a}|f(x)|dx = 0, then f(x) = 0 for all x \in [a,b] [/itex]

My Answer: EDIT: This is True.

Correct answers?

Thanks for checking in with me and guiding me.

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