Calculus I - Area Between Curves - Mistake on Answer Key

[GreenPrint]

This is the answer key to one of my quizzes

If you notice in the question, see attachment, were it says to integrate with respect to y the integral is integrating from 0 to 25 but this produces a negative area so this is technically wrong, yes? You don't just simply integrate from the lower value to the upper value for all cases? Like in this problem the proper integral would of been from 25 to 0 were your actually integrating from the higher value to the lower value... the reasoning behind this is because the higher valued limit appear to the left of the lower value limit 0 so the x coordinate of the higher value limit is -5 which is less than the x coordinate of the lower valued limit 0

the points of intersection of the functions are:
(-5,25)(0,0)
so is my reasoning correct for as to why you actually would integrate from 25 to 0 instead of 0 to 25 to produce a positive area?

 

[Mark44]

This is the answer key to one of my quizzes

If you notice in the question, see attachment, were it says to integrate with respect to y the integral is integrating from 0 to 25 but this produces a negative area so this is technically wrong, yes?

No. Integrating from 0 to 25 produces a positive number, 125/6, the same as the other integral.

You don't just simply integrate from the lower value to the upper value for all cases? Like in this problem the proper integral would of been from 25 to 0 were your actually integrating from the higher value to the lower value... the reasoning behind this is because the higher valued limit appear to the left of the lower value limit 0 so the x coordinate of the higher value limit is -5 which is less than the x coordinate of the lower valued limit 0

the points of intersection of the functions are:
(-5,25)(0,0)
so is my reasoning correct for as to why you actually would integrate from 25 to 0 instead of 0 to 25 to produce a positive area?

 

[ArcanaNoir]

I got a positive area doing it the way it's written on the answer key.

 

[GreenPrint]

Hm I guess I entered it into my calculator wrong thanks, so it's always the integral from the lower value to the higher value after all than?

 

[Mark44]

Yes, if you are using vertical strips. The area of one of these vertical strips is (<upper y value> - <lower y value>) * dx.

If you're using horizontal strips, the area of a strip is (<right x value> - <left x value>) * dy.