Stuck finding area under the curve

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October 2nd 2010, 08:42 AM
solidstatemath

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Stuck finding area under the curve

I'm stuck at the question 51 in my textbook.

I can find the two equations and know the f(x)=y=1 is greater than g(x). I follow the solutions and get stuck at the part where they go from 1 - cos^2 x = sin^2 x.

1 - cos^2 x makes sense to me since it's f(x) - g(x) but where does the sin^2 x come from. And from this point on I don't understand why they divide it by 2 and create the integral off the interval.

One attachment has the textbook question, #51 and the other is my solutions book scan.
October 2nd 2010, 08:55 AM
skeeter

Quote:

Originally Posted by solidstatemath

I'm stuck at the question 51 in my textbook.

I can find the two equations and know the f(x)=y=1 is greater than g(x). I follow the solutions and get stuck at the part where they go from 1 - cos^2 x = sin^2 x.

1 - cos^2 x makes sense to me since it's f(x) - g(x) but where does the sin^2 x come from. And from this point on I don't understand why they divide it by 2 and create the integral off the interval.

One attachment has the textbook question, #51 and the other is my solutions book scan.

being a calculus student, you should be familiar with both of these identities ...

$\sin^2{x} + \cos^2{x} = 1$

and ...

$\displaystyle \sin^2{x} = \frac{1-\cos(2x)}{2}$
October 2nd 2010, 08:56 AM
Prove It

The Pythagorean Identity states that

$\cos^2{x} + \sin^2{x} = 1$ .

Thus $1 - \cos^2{x} = \sin^2{x}$ .

You should also know the double angle identity

$\cos{2x} = \cos^2{x} - \sin^2{x}$ .

A combination of the double angle identity and the Pythagorean Identity gives

$\sin^2{x} = \frac{1 - \cos{2x}}{2}$ .

They do this conversion to simplify the problem.
October 2nd 2010, 08:58 AM
skeeter

Quote:

Originally Posted by Prove It

The Pythagorean Identity states that

$\cos^2{x} + \sin^2{x} = 1$ .

Thus $1 - \cos^2{x} = \sin^2{x}$ .

They do this conversion to simplify the problem.

fify
October 2nd 2010, 08:59 AM
Prove It

Quote:

Originally Posted by skeeter

fify

What does fify mean?
October 2nd 2010, 09:02 AM
skeeter

Quote:

Originally Posted by Prove It

What does fify mean?

fixed it for you
October 2nd 2010, 09:03 AM
Prove It

Quote:

Originally Posted by skeeter

fixed it for you

I see, thank you. Serves me right for trying to do this at 4am hahaha.
October 2nd 2010, 09:12 AM
sunmalus

$Sin(x)^2+ Cos(x)^2=1$ if you think about it for a sec it is Pythagoras theorem. So $Sin(x)^2+ Cos(x)^2=1 \Rightarrow Sin(x)^2=1-Cos(x)^2$ . For the next step you can use $Sin(\alpha)Sin(\beta)=\frac{1}{2}((-Cos(\alpha + \beta)+Cos(\alpha -\beta)))$ with alpha=beta and that's it.