Thread: Stuck finding area under the curve

1. 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.

2. 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 ...

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

3. 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.

4. 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

5. Originally Posted by skeeter
fify
What does fify mean?

6. Originally Posted by Prove It
What does fify mean?
fixed it for you

7. Originally Posted by skeeter
fixed it for you
I see, thank you. Serves me right for trying to do this at 4am hahaha.

8. $\displaystyle Sin(x)^2+ Cos(x)^2=1$ 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.