Not Just Another Sine Integral

Calculus Level 5

In the picture above, the graph of $|y|\leq \sin(x)$ is super imposed upon the human eye.

Now, assume that the human eye is modeled by the function $|y|\leq \sin(x)$ where $x$ is in $\large{\text{degrees}}$ and $0\leq x \leq 180$

Find the area of a human eye to two decimal places.

The answer is 229.18.

1 solution

Trevor Arashiro
Mar 18, 2015

The solution is quite simple, all we have to do is convert $\sin(x)$ from degrees to radians. Also, since the graph is symmetric about the x axis, all we have to do is take the area above the curve and multiply it by 2.

$\displaystyle \int_0^{\pi} \sin(x)~\Bbb{d}x=\displaystyle \int_0^{180} \sin \left(\frac{\pi x^{\circ}}{180}\right)~\Bbb{d}x~~ \text{(note the degree sign)}$

Let $u=\dfrac{\pi x}{180}$ and $\dfrac{180\Bbb{ d}u}{\pi}=\Bbb{d}x$

$\frac{180}{\pi}\displaystyle \int_0^{180} \sin (u)~\Bbb{d}u$

$\frac{180}{\pi}\left[\cos(0)-\cos\left(\frac{\pi(180)}{180}\right)\right]$

$\dfrac{180}{\pi}[2]$

Multiplying through by 2 to account for both sides we get

$\boxed{\dfrac{720}{\pi}\approx 229.18~}$

Huh, this made my head spin a bit, but I got it on my second attempt. What I would want is an integral calculator that gives the same answer for

$\displaystyle\int_{0}^{180} \sin(x) dx$ in degree mode and $\displaystyle\int_{0}^{\pi} \sin(x) dx$ in radian mode.

Then I wouldn't have to, you know, think. :P

P.S.. I guess the units would be mm $^{2}$ .

Brian Charlesworth - 6 years, 2 months ago

Is it $\frac{720}{\pi}$ ? 'coz it has to account for both side.

คลุง แจ็ค - 6 years, 2 months ago

@Trevor Arashiro , I think it should be $\dfrac{720}{\pi}$ , which is $\boxed{\approx 229.18 ~}$

Vishwak Srinivasan - 6 years, 2 months ago

Oh, thanks, nice catch

Trevor Arashiro - 6 years, 2 months ago

Not Just Another Sine Integral

The answer is 229.18.

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