How do I know which function to subtract when finding areas enclosed by curves?

Strat

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I keep getting the signs wrong on my answers when I try to find the area enclosed by two curves. I know you integrate from the lowest x or lowest y to the highest so I think I am subtracting the wrong function.

When the function is with respect to x (like in y=x+1), I thought you were supposed to take the rightmost function and subtract the leftmost. Is this correct?

When the function is with respect to y (like in x = y+1), are you supposed to subtract the lower function from the upper function?

I'm not sure how to tell which is upper or rightmost other than by eyeing it. Is there a more rigorous way?

Find the area of the region between the functions x = 2y2 and x = 4 + y2 . I thought x=2y2 was the rightmost function. But after I integrated I got -32/3 when I should have gotten 32/3.
 
I would assume that the first thing you did was check for the x-limits of integration. The two graphs cross when x= 2y^2= y^2+ 4. That gives 2y^2- (y^2+ 4)= y^2- 4= (y+ 2)(y- 2)= 0 so y= -2 and 2.

That should also show you that 2y^2- (y^2+ 4)= y^2- 4= (y- 2)(y+ 2)= (y- 2)(y- (-2)). Between -2 and 2, y+ 2 is positive while y- 2 is negative. The product, so 2y^2- (y^2+ 4) is negative. That means that 2y^2 is less than y^2+ 4 between -2 and 2.
 
I keep getting the signs wrong on my answers when I try to find the area enclosed by two curves. I know you integrate from the lowest x or lowest y to the highest so I think I am subtracting the wrong function.

When the function is with respect to x (like in y=x+1), I thought you were supposed to take the rightmost function and subtract the leftmost. Is this correct?
When you did Riemann sums, where you finding the areas of rectangles of fixed heights but widths that varied according to the function(s), or the other way around? Do the same for integration. (Check the worked examples in your book, where they always subtract the lower function - or x-axis - from the upper function, if you're not sure.)

When the function is with respect to y (like in x = y+1), are you supposed to subtract the lower function from the upper function?
You turn the graph on its side, and proceed as usual, just using "y" instead of "x".

I'm not sure how to tell which is upper or rightmost other than by eyeing it. Is there a more rigorous way?
Look at the graph. The line that is higher is the upper function; the other one is the lower function.

Find the area of the region between the functions x = 2y2 and x = 4 + y2 . I thought x=2y2 was the rightmost function. But after I integrated I got -32/3 when I should have gotten 32/3.
Not being able to see your steps, we cannot comment on where things may have gone wrong. However, a quick graph should suffice to show which graph is "above" the other (on the sideways graph). Then set the two equations equal to find the intersection points, being at y = +2, and y = -2. The first graph is "below" the other one, since they're both positive quadratics, but the second graph has a vertex four units higher than the other graph.

What did you see, when you graphed the two curves?

Please be complete. Thank you! ;)
 
@Stapel, @HallsofIvy

Thanks, I think I've cleared up how to think of which is the higher function.
 
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