Total area with integration and negative signs

[BakerDad12]

Hey, I'm confused about finding the area under a curve. If we are asked to find the area under a curve and the curve goes under the x-axis and above the x-axis, we would get a negative area and a positive area. I've attached a picture below to show what I mean.

Our textbook says after taking the definite integral, to take the absolute value of both bounds - essentially ignore the negative signs, if there are any. So we are trying to find the total magnitude I suppose. But is this correct? I'm just wondering because I haven't seen it anywhere else and the textbook has been wrong before.

[Einstein_Reborn_97]

Yes, this is correct. The definite integral gives the signed area under a curve. To find areas below the x-axis, take the absolute value of the definite integral. If the word "area" is mentioned/implied in the question, use absolute values. If not, i.e. "definite integral" is used, then don't.

[djaco75]

Hey, I'm confused about finding the area under a curve. If we are asked to find the area under a curve and the curve goes under the x-axis and above the x-axis, we would get a negative area and a positive area. I've attached a picture below to show what I mean.

Our textbook says after taking the definite integral, to take the absolute value of both bounds - essentially ignore the negative signs, if there are any. So we are trying to find the total magnitude I suppose. But is this correct? I'm just wondering because I haven't seen it anywhere else and the textbook has been wrong before.

my teacher always says:
-if the question asks to 'evaluate the definite integral' or just has the word evaluate, just shove the numbers in
-if it asks to 'find the area under the curve' you need to take into account signed areas

[Einstein_Reborn_97]

my teacher always says:
-if the question asks to 'evaluate the definite integral' or just has the word evaluate, just shove the numbers in
-if it asks to 'find the area under the curve' you need to take into account signed areas

Your teacher is right.