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3 years ago

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A circle with radius of 2cm sits inside a 7cm x 11cm rectangle. What is the area of the shaded region? Round your final answer t

o the nearest hundredth.

2 answers:

Novosadov [1.4K]3 years ago

4 0

The solution is attached as a word file

Mama L [17]3 years ago

3 0

Answer:

64.43 cm^2

Step-by-step explanation:

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Split $C$ into two component segments, $C_1$ and $C_2$ , parameterized by

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respectively, with $0\le t\le1$ , where $\mathbf r_i(t)=(x(t),y(t))$ .

We have

$\mathrm d\mathbf r_1=(6,1)\,\mathrm dt$

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where $\mathrm d\mathbf r_i=\left(\dfrac{\mathrm dx}{\mathrm dt},\dfrac{\mathrm dy}{\mathrm dt}\right)\,\mathrm dt$

so the line integral becomes

$\displaystyle\int_C(x+6y)\,\mathrm dx+x^2\,\mathrm dy=\left\{\int_{C_1}+\int_{C_2}\right\}(x+6y,x^2)\cdot(\mathrm dx,\mathrm dy)$

$=\displaystyle\int_0^1(6t+6t,(6t)^2)\cdot(6,1)\,\mathrm dt+\int_0^1((6+t)+6(1-t),(6+t)^2)\cdot(1,-1)\,\mathrm dt$

$=\displaystyle\int_0^1(35t^2+55t-24)\,\mathrm dt=\frac{91}6$

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