# Stuck finding area under the curve

Printable View

• Oct 2nd 2010, 08:42 AM
solidstatemath
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.
• Oct 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 ...

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

and ...

$\displaystyle \displaystyle \sin^2{x} = \frac{1-\cos(2x)}{2}$
• Oct 2nd 2010, 08:56 AM
Prove It
The Pythagorean Identity states that

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

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

You should also know the double angle identity

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

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

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

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

Originally Posted by Prove It
The Pythagorean Identity states that

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

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

They do this conversion to simplify the problem.

fify
• Oct 2nd 2010, 08:59 AM
Prove It
Quote:

Originally Posted by skeeter
fify

What does fify mean?
• Oct 2nd 2010, 09:02 AM
skeeter
Quote:

Originally Posted by Prove It
What does fify mean?

fixed it for you
• Oct 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.
• Oct 2nd 2010, 09:12 AM
sunmalus
$\displaystyle Sin(x)^2+ Cos(x)^2=1$ if you think about it for a sec it is Pythagoras theorem. So $\displaystyle Sin(x)^2+ Cos(x)^2=1 \Rightarrow Sin(x)^2=1-Cos(x)^2$. For the next step you can use $\displaystyle Sin(\alpha)Sin(\beta)=\frac{1}{2}((-Cos(\alpha + \beta)+Cos(\alpha -\beta)))$ with alpha=beta and that's it.