3 years ago

Find the area of the shaded region. Round your answer to nearest tenth.

1 answer:

6 0

Answer:

About 17.38 m

Step-by-step explanation:

Radius of the quarter circles are 4.5

pi*r^2

pi*4.5^2=63.6172512352

81-63.6172512352=17.3827487648

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$f(x)=3\sin x+3\cos x\implies f'(x)=3\cos x-3\sin x$

$f$ has critical points where $f'=0$ :

$3\cos x-3\sin x=0\implies\cos x=\sin x\implies\tan x=1\implies x=\pm\dfrac\pi4+2n\pi$

where $n$ is any integer. We get solutions in the interval $\left[0,\frac\pi3\right]$ for $n=0$ , for which $x=\frac\pi4$ .

At this critical point, we have $f\left(\frac\pi4\right)=3\sqrt2\approx4.243$ .

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$\max\limits_{x\in\left[0,\frac\pi3\right]}f=3\sqrt2$

$\min\limits_{x\in\left[0,\frac\pi3\right]}f=3$

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