gilded shoal
I am kinda confused.
when we find the definite integral with limits a and b for a graph that lies entirely below the x axis (i have chosen a negative quadratic for no apparent reason) are we calculating the green area or the red area, and furthermore since the red area is negative, would taking the absolute value of this equate it to the green area?
I asked wolfram and got almost different answers every time in different chats, and im confused asf
torn impBOT
gilded shoal
am i even right in thinking the area calculated by using the definite integral with limits is the area UNDER the curve?
if so, the red area seems like it would be the right area we are calculating
furthermore for a sine graph, how come we calculate this blue area instead of calculating this green area, if the integral gives us the area UNDER the curve
wind briar
I am kinda confused.
when we find the definite integral with limits a and b for ...
Well, if the function f is strictly negative, you can take -f, which is strictly positive. Since -int(-f(x)dx) is the same as int(f(x)dx), the integral of f is basically the integral of -f but flipped, so instead of thinking about area underneath the curve, think about area between the curve and the x axis
The integral flips alongside f, basically
Does that help you answer the question yourself?
gilded shoal
Does that help you answer the question yourself?
it does kinda
but does f flipping mean that we get the green area instead of the red area
like are they equivalent in any way?
gilded shoal
Well, if the function f is strictly negative, you can take -f, which is strictly...
do we only consider the area under the curves for functions for which f(x) has a positive region asw, and then for strictly negative functions we consider the area between the curve and the x axis, AKA above the curve?
because for a postitive parabola below the x axis like this
we can only calculate the area between the above the curve and the x axis right?
if we were to do it below then we would get an infinite area right, because despite the limits it doesnt really give us an indication of where the area would stop, is this correct?
<@&286206848099549185>
topaz tinsel
do we only consider the area under the curves for functions for which f(x) has a...
Yeah, you care about the area bounded by the function f(x) and the x-axis. Any area below the axis is negative and above is positive.
gilded shoal
Yeah, you care about the area bounded by the function f(x) and the x-axis. Any a...
so is this wrong
topaz tinsel
Yes, this is wrong. You wouldn't get this with that integral.
gilded shoal
is this yellow area what we would find
topaz tinsel
Yes
gilded shoal
ah so its not actually the area UNDER the curve?
topaz tinsel
You could find the red area, but you'd have to use the y-axis and some tricks.
topaz tinsel
ah so its not actually the area UNDER the curve?
I imagine it like the area between curve and axis.
gilded shoal
You could find the red area, but you'd have to use the y-axis and some tricks.
so we would never be asked to find the red area in edexcel a level maths?
gilded shoal
I imagine it like the area between curve and axis.
thank you so much, wolfram was telling me im right one second and wrong the next second
topaz tinsel
so we would never be asked to find the red area in edexcel a level maths?
not sure, how in deep your course goes, but I dealt with similar problems in the American AP Calculus BC course.
So you might get to the red area problems, but I assume that'll take a while.
topaz tinsel
thank you so much, wolfram was telling me im right one second and wrong the next...
no worries, I had similar problems a year ago, when I was studying it ^^
gilded shoal
no worries, I had similar problems a year ago, when I was studying it ^^
okay for sure thanks again bro
i guess if they wanted us to find the area under the curve for a strictly negative function they would give us some typa visual
in the textbook but as far as i know they havent
topaz tinsel
I mean for a parabola you'd still be able to find the area without any nasty tricks. At least I think
topaz tinsel
Does this mean f(a)=f(b)?
gilded shoal
Does this mean f(a)=f(b)?
no its, just a negative parabola f(x) where f(x)<0
just shading areas for visual clarity
topaz tinsel
Ah, so the dashed line is just for visuals?
Well that's a shame I wanted to show a trick, where you could find the red area.
gilded shoal
Well that's a shame I wanted to show a trick, where you could find the red area.
im still open to learning it ngl
more maths the better
gilded shoal
Ah, so the dashed line is just for visuals?
yeah just to show the region of areas
that i wanted to know were right
yellow is correct as you've stated, and red is wrong when just using stanadrd definite integrals right?
topaz tinsel
If you had f(a)=f(b), then you'd get a rectangle. You can find the area of the rectangle using f(a or b) * (b - a). If you the curve stays in the rectangle then you can say that rectangle area + integral from a to b would give you the red area.
or in a more natural language: rectangle - yellow area = red area.
Right?
gilded shoal
If you had f(a)=f(b), then you'd get a rectangle. You can find the area of the r...
also im sorry for being so stubborn about this but
for integral with limits a and b, or, c to d, we would get green regions right
and for b to c we would get blue region?
topaz tinsel
right
topaz tinsel
that isn't stubborness, that's a normal question :D
topaz tinsel
so thats correctr?
yessir
gilded shoal
that isn't stubborness, that's a normal question :D
i appreciate that, i didnt wanna annoy you by asking essentially the same thing like 5 times 😭
i just really wanna make sure i understand this
topaz tinsel
no no go ahead.
gilded shoal
would you mind if i added you as a friend and asked you questions in dms?
in the future
topaz tinsel
for integral with limits a and b, or, c to d, we would get green regions right
This also import. If you try to find this area you might say the total area = the absolute value of the integral, but in this case that might not be true. Always be careful when function crosses the x-axis.
topaz tinsel
would you mind if i added you as a friend and asked you questions in dms?
sure sure, but I'm often busy. Try it if you want, but if I'm busy just try messaging here. I trust there's a lot of helpful people here too.
gilded shoal
sure sure, but I'm often busy. Try it if you want, but if I'm busy just try mess...
for sure bro, i wont bother you i will ask here
but thank you again
i started sweating cus i didnt understand this fr
wolfram is both a gift and a curse
topaz tinsel
I mean go ahead, I'll gladly help, it's just there's a lot of stuff coming up this month, so might not answer asap.
gilded shoal
thanks brodi, i really appreciate it
topaz tinsel
I'll mark this problem as solved if you don't mind.
.solved
oh wait maybe you're supposed to do it... oops
gilded shoal
oh wait maybe you're supposed to do it... oops
oh yeah my bad ill do it rn
gilded shoal
oh wait maybe you're supposed to do it... oops
how do i do it? 😅
gilded shoal
.solved
torn impBOT
Solved
gilded shoal
nice bro than you
