Calculating the Area Under a Trig Graph

[Miike012]

Question: Find the area under the graph y = cos(x) from x = 0 to x = pi/2

Solution:

A = Lim ( ∏/(2n) * Ʃ cos( ∏i/(2n)) = ? Start: i = 0 and End: n = n
n → ∞

Just like there is a theorem for adding consecutive numbers... n(n + 1)/2..
Is there one for trig functions?

 

[LCKurtz]

I don't know of any nice formula for that sum. (That doesn't mean there isn't one.) Are you studying approximating sums for integrals but don't have the fundamental theorem of calculus yet?

 

[Miike012]

I guess you can say that... The real question was ... estimate the area of cos(x) [0,pi/2] using 4 approximating rectangles and right endpoints..

That is easy so I wanted to try and solve for n rectangles..

 

[LCKurtz]

I guess you can say that... The real question was ... estimate the area of cos(x) [0,pi/2] using 4 approximating rectangles and right endpoints..

That is easy so I wanted to try and solve for n rectangles..

That's good that you found it so easy. That means you get the idea and that is what counts. There is a reason calculus books typically only do the limit thing for parabolas. Try that if for y = x² if your book hasn't already done it for you. You will be able to calculate it.

Trying it for most functions will leave you with a sum that you can't evaluate in closed form, such as you have just experienced. That is why the fundamental theorem of calculus is so important.

 

[Miike012]

one more question...

Estimate the area under the graph of f(x) = 1 + x^2 from x = -1 to x = 2 using 6 rectangles and right end point.

Question:
Can I change the equation from 1 + x^2 to 1 + (x - 1)^2 and change the interval to x = 0 to x = 3 ??

This seems logical because technically it would be the same area.. and it is easier for me to break up into 6 rectangles.

 

[Dick]

I don't know of any nice formula for that sum. (That doesn't mean there isn't one.) Are you studying approximating sums for integrals but don't have the fundamental theorem of calculus yet?

Just for the record, there are formulas like that. See http://en.wikipedia.org/wiki/List_of_trigonometric_identities Look under "Other sums of trigonometric functions". You can derive them by summing the geometric series exp(i*a*k) and splitting into real and imaginary parts.

 

[LCKurtz]

one more question...

Estimate the area under the graph of f(x) = 1 + x^2 from x = -1 to x = 2 using 6 rectangles and right end point.

Question:
Can I change the equation from 1 + x^2 to 1 + (x - 1)^2 and change the interval to x = 0 to x = 3 ??

This seems logical because technically it would be the same area.. and it is easier for me to break up into 6 rectangles.

Yes you could, but I don't see why it is any easier.