Professor Organic Chemistry Tutor, this is an outstanding analysis and derivation of the Area of an Ellipse in Calculus Two. There are other methods/ways to derive the Area of an Ellipse in Calculus. This is an error free video/lecture on RUclips TV with the Organic Chemistry Tutor.
I know a derivation of the area with greens theorem is possible - plus, there is a definition of greens theorem using a double integral, hence both should be possible
No, that's the right way. The reason why you become a negative value is that when you integrate over the first quadrant, you need to take the absolute value of the function. So when you you derive acos(x) = -asin(x) --> and you want to bring out the constants out of the integral, the '-' disappears. Regards
Lets say the circle is a speacial case of ellipse where a=b=r so area becomes a*b*pi=pi*r*r*, now lets say i remove some portion of length from a and give it to b, so now it is still an ellpise but now the area should become (r-x)(r+x)*pi=(r^2-x^2)*pi which is not the same as pi*r^2 as before.
iconsider also (r-x)(r+y)^pi=A if we define a function x=ky or x=y-20 lets say, (what this means is that im removing some portion of length from a but im not giving it fully to b or manipulating it and then giving it to b) then it might be possible to get the same area idk this is just a random guess
I know this is an old comment, but he did this because the integral would only be the area of one quadrant (in this case, the first) of the ellipse, so he multiplied by four to account for all of them
Look. Sine function has domain ranging from minus infinity to plus infinity. So whatever real value of x and a you have, you can always have a value of theta such that x equals a sin of theta. It's just for simplification of the integral so it's easier to find the integral.
Hi, can you please cover rotating parabola and general equation of parabola, i cant find a good explanation video on them on youtube. Also can you cover the directrix, and latus rectum of ellipse, will be qiite helpful! Pove your videos as always
I was never taught that the square root of (a square times (whatever number)) is the same as the suare root of (a square) times (whatever number). I can't wrap my head around it. How was I supposed to solve this problem without that info?
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Professor Organic Chemistry Tutor, this is an outstanding analysis and derivation of the Area of an Ellipse in Calculus Two. There are other methods/ways to derive the Area of an Ellipse in Calculus. This is an error free video/lecture on RUclips TV with the Organic Chemistry Tutor.
Amm, I think [1-(x/a)^2] should be under root too?
that is exactly what I'm thinking
Isaiah Burkes he fixed it at 5:36
Could you also do so using a double integral and Green's Theorem?
I know a derivation of the area with greens theorem is possible - plus, there is a definition of greens theorem using a double integral, hence both should be possible
What a beautiful derivation
The integral of Sqrt(a^2-x^2) under 0 to a is pi*a^2/4
I used x=acosØ instead of x=sinØ and got -πab, but I don't know why that is wrong. Can someone help me out?
No, that's the right way. The reason why you become a negative value is that when you integrate over the first quadrant, you need to take the absolute value of the function. So when you you derive acos(x) = -asin(x) --> and you want to bring out the constants out of the integral, the '-' disappears. Regards
got a question, If you squash a circle into an ellipse, will the area remain the same?
Lets say the circle is a speacial case of ellipse where a=b=r so area becomes a*b*pi=pi*r*r*, now lets say i remove some portion of length from a and give it to b, so now it is still an ellpise but now the area should become (r-x)(r+x)*pi=(r^2-x^2)*pi which is not the same as pi*r^2 as before.
iconsider also (r-x)(r+y)^pi=A
if we define a function x=ky or x=y-20 lets say, (what this means is that im removing some portion of length from a but im not giving it fully to b or manipulating it and then giving it to b) then it might be possible to get the same area idk this is just a random guess
Clearly not always. When you squash one axis to zero the area goes to zero. Just a thought
Why x can become asin(θ)?
Why did you multiply your integral by 4 ??
I know this is an old comment, but he did this because the integral would only be the area of one quadrant (in this case, the first) of the ellipse, so he multiplied by four to account for all of them
@@ry4n.bg6thanksss
@ry4n.bg6 thanks mate. Ur cmnt helped me ❤💪
@@ry4n.bg6wasnt he trying to find the area of one quadrant though
This guy literally knows everything 😂
weird thing is if i multiple the constant 2 inside the integral or separate the intergral ,i will get 2abπ
Where did you get that x=asin(theta)
Look. Sine function has domain ranging from minus infinity to plus infinity. So whatever real value of x and a you have, you can always have a value of theta such that x equals a sin of theta. It's just for simplification of the integral so it's easier to find the integral.
Thank you for this. Very useful!
Hi, can you please cover rotating parabola and general equation of parabola, i cant find a good explanation video on them on youtube. Also can you cover the directrix, and latus rectum of ellipse, will be qiite helpful!
Pove your videos as always
lovely video thank you
man that is very good but the sound is so low
please turn it up
thanks
Try solving this with beta function its more easy and quick that way
I was never taught that the square root of (a square times (whatever number)) is the same as the suare root of (a square) times (whatever number). I can't wrap my head around it. How was I supposed to solve this problem without that info?
He made a mistake at that point, watch a little bit forward :D
Thank you
You could have just swapped the limits with respect to theta, 0->a => 0->π/2
How quickly these videos are uploaded?
I am happy but its too fast.
1909 lets play pinball 💘💘
💕💕💕💕
Easily integration could be done bro