Integrating Trigonometric Functions: How to Remember and Use Identities?

athrun200 · Jul 17, 2011

Homework Statement

I have a problem in part b

Printed textbook problem 10: a uniform chain hangs in the shape of the catenary y = cosh x between x = -1 and x = 1. Find (a) its length, (b) y-bar, with part (b) circled.

Homework Equations

The Attempt at a Solution

How to integral the function?

Handwritten derivation on lined paper: arc length l = integral of ds, reduced to integral from -1 to 1 of cosh x dx giving 2 sinh 1, then part (b) integral of y-bar rho ds = integral of y rho ds leading to integral from -1 to 1 of cos^2 hx dx.

athrun200 · Jul 17, 2011

1. Homework Statement
I would like to ask for 18 and20

Printed textbook page listing calculus problems 17 to 31 about the curve y = sqrt(x) between x = 0 and x = 2, with problems 18 and 20 circled.

2. Homework Equations

3. The Attempt at a Solution
Again how to integral the function?

Handwritten calculus working on lined paper: arc length l = integral of ds, equal to the integral from 0 to 2 of the square root of 1 + 1/(4x) dx, then a limit as a tends to infinity of the integral from a to root 8 of u^2 over 2(u^2-1)^2 du, with the substitution u = root(1 + 1/(4x)) and its derivatives beside it.

Handwritten surface area setup on lined paper labelled 20: S = integral from 0 to 2 of 2πy ds, then 2π times integral from 0 to 2 of sqrt(x) sqrt(1 + 1/(4x)) dx, then 2π times integral from 0 to 2 of sqrt(x + 1/4) dx.

gb7nash · Jul 17, 2011

10a looks fine.

For 10b, take advantage of the following trig identity:

cosh² x = [1 + cosh (2x)]/2

athrun200 · Jul 17, 2011

gb7nash said:

10a looks fine.

For 10b, take advantage of the following trig identity:

cosh² x = [1 + cosh (2x)]/2

Oh no, I know this identity, but there are too many identity and formula so I can't remember it before you tell me. How can I make sure that I remember all of them? Or where can I check these useful identity?

gb7nash · Jul 17, 2011

http://planetmath.org/encyclopedia/HyperbolicIdentities.html

As far as knowing which ones to remember, it would depend on what you're trying to do and what class you're in. For a lot of integral problems, the identities I usually see are sin² + cos² = 1 (and any other pythagorean identities), and half/double angle formulas.

athrun200 · Jul 17, 2011

gb7nash said:

http://planetmath.org/encyclopedia/HyperbolicIdentities.html

As far as knowing which ones to remember, it would depend on what you're trying to do and what class you're in. For a lot of integral problems, the identities I usually see are sin² + cos² = 1 (and any other pythagorean identities), and half/double angle formulas.

Thanks so much.
Shall we move on #2?
I just have some problems.

Integrating Trigonometric Functions: How to Remember and Use Identities?

Homework Statement

Homework Equations

The Attempt at a Solution

Discussion

Similar threads