Question

AP Calc Integration question: Some questions that ask you to calculate the area between the 2 graphs don't give me the pictures of the graphs. Therefore, it's hard to determine which of the functions is the upper function and which is the lower function without a graph. Is there a way to calculate with function is upper and which is lower without using a graphing calculator?

Answer 1

I'd say just plug in the endpoints and a point in the middle to get a general sense of what's going on

Answer 2

Find the intersections.
Understand the shape of the graphs in that Domain.
Test one point in any region.

Answer 3

How do I "understand the shape of the graphs in that domain"?
Thank you for your answers by the way!

Answer 4

why would you need to determine the top and bottom functions?

Answer 5

the area between curves will always be postiive, so if you get a negative number thats just because you subtracted the top from the bottom, but the only thing that changes is sign.

for example:

5-2 = 3
2-5 = -3

Area doesnt care about direction, so the results is 3 regardless.

Answer 6

Oh!

Answer 7

so you just change the sign after, if the area turns out negative? @amistre64

Answer 8

That works right up until you have both positive and negative in the same integral. You must find ALL the intersections and treat the regions separately.

"Understand the graphs" is of limited usefulness. Some are more understandable than others. You should know what a line does, or a parabola, for example.

Answer 9

Ok, @tkhunny , so you are saying that @amistre64 's method works, but you just need to divide up the area so that positive and negative are not in the same integral?

Answer 10

But then to divide up the area, I would need the picture of the graphs again..so wouldn't that make the method obsolete?

Answer 11

No, you must be very careful to find ALL intersections. All of them. If you miss one, you're done.

This is also what "understand the functions" means. If you know something is monotonic increasing, well, it never comes back down - for example. It helps to know where the first derivatives change sign.

Answer 12

are your functions defined? or are you just not able to see the graphs of them?

Answer 13

Excellent point. This is another part of "understand the functions".

Answer 14

yes they are defined...for example e^(x/2)

Answer 15

@amistre64 please continue :)

Answer 16

if the functions cross within the interval, then you will need to determine that intersection point, and divide the integration between them

Answer 17

the area between top and bottom will always be positive, but you still need to know where they intersect otherwise your sum is going to flip over and your going to start subtracting.

the area between curves is different from taking an integration from a to b of a function, which is why this process results in negative values.

Answer 18

wht are the defined functions within the interval? and what is your defined interval?

Answer 19

For the picture you drew, @amistre64 , which parts are the parts that I need to subtract?

Answer 20

a) y = 2sin(x), and y = 2cos(x) from pi/4 to 5pi/4

is this one question?

or is b part of it??

Answer 21

they are separate questions

Answer 22

well, when does 2sin(x) = 2cos(x) ?

Answer 23

um..

Answer 24

the 2s are irrelevant, when is sin(x) = cos(x)

Answer 25

pi/4

Answer 26

is that right?

Answer 27

yes, and also at 5pi/4 and that is the only points in the domain that these function cross.