# Thread: integration area of sinx and cos x

1. ## integration area of sinx and cos x

So, How does one find the total shaded area (red and green in the attachment).
I'm unsure whether the first half (red part) is going to be the integral of sinx or (cosx -sinx)

My instinct working out for total area of the shaded:

integration of sin x between 0 and 45 + [integration of sinx -cos x in between 45 to 90]

2. ## Re: integration area of sinx and cos x

Originally Posted by cloud edwards
So, How does one find the total shaded area (red and green in the attachment).
I'm unsure whether the first half (red part) is going to be the integral of sinx or (cosx -sinx)

My instinct working out for total area of the shaded:

integration of sin x between 0 and 45 + [integration of sinx -cos x in between 45 to 90]
$\noindent A = \int_0^{\frac{\pi}{4}}\sin{x}\mathrm{d}x + \int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\cos{x}\mathrm {d}x = 2\int_0^{\frac{\pi}{4}}\sin{x}\mathrm{d}x (by symmetry). And so A = (2 - \sqrt{2}) units^2$

3. ## Re: integration area of sinx and cos x

Ok thanx, so how did u get the second half (i.e. cosx)

4. ## Re: integration area of sinx and cos x

Originally Posted by cloud edwards
Ok thanx, so how did u get the second half (i.e. cosx)
$\noindent The green shaded area is below the curve y = \cos{x} and is not bounded by the curve y = \sin{x}. Instead, the area is bounded by the line x = \frac{\pi}{4}, the curve y = \cos{x} and the x-axis.$

5. ## Re: integration area of sinx and cos x

Thanx bro, you're a lifesaver!

6. ## Re: integration area of sinx and cos x

Going to be a bit pedantic here, but you shouldn't be using your limits in terms of degree but in radians, it could create a lot of awkward scaling factors especially if you're combining trig functions with some linear function.

e.g $\int_0^{45^{\circ}} \ \ sinx - x \ dx$

wouldn't give you the right value for the area under the curve.

7. ## Re: integration area of sinx and cos x

How different would it be if i didn't use radians, i'm a bit confused with the example,
sorry.

8. ## Re: integration area of sinx and cos x

Originally Posted by cloud edwards
How different would it be if i didn't use radians, i'm a bit confused with the example,
sorry.
It is how the derivative of cos x and sin x is defined. We use radians instead of degrees as the computation is simpler.
for x in degrees
$\frac{d}{dx} \sin{x} = \frac{\pi}{180}\cos{x}$

9. ## Re: integration area of sinx and cos x

Ok sweet, thanx guys

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