Area common to two regions

Consider the regions $(x,y)|\hspace{2mm}x^{2}+y^{2}\leq100$ and $sin(x+y)<0$ .

Find the area of the region common to the above inequalities.

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

If $A_1 = \{(x,y)\,|\, x^2+y^2 \le 100 , \sin(x+y) > 0\}$ and $A_2 = \{(x,y) \,|\, x^2+y^2 \le 100: \sin(x+y) < 0\}$ , then a rotation through $\pi$ about the origin maps $A_1$ onto $A_2$ , and vice versa. Thus $A_1$ and $A_2$ have the same area. The sets are disjoint, and their union is the interior of the circle (with $9$ lines removed), and so the area of each is $\tfrac12 \times 10^2\pi = 50\pi$ .

Mark Hennings - 7 years, 8 months ago

Nicely Done Mark !!

Gabriel Merces - 7 years, 8 months ago

Great work!!

Piyal De - 7 years, 8 months ago

Hello Mark!

What does this statement mean: "then a rotation through $π$ about the origin maps $A_1$ onto $A_2$ , and vice versa."? Can you please post a link which explains this?

Thanks!

Pranav Arora - 7 years, 8 months ago

The point $(x,y)$ belongs to $A_1$ if and only if the point $(-x,-y)$ belongs to $A_2$ . Changing the sign of both $x$ and $y$ is achieved by a rotation about the origin through $\pi$ (or successive reflections in the $x$ - and $y$ -axes, or an enlargement scale factor $-1$ through the origin). However you describe this transformation of the plane $\mathbb{R}^2$ , it preserves areas, so that $A_1$ and $A_2$ have the same area.

Mark Hennings - 7 years, 8 months ago

@Mark Hennings – Thanks Mark! :)

Pranav Arora - 7 years, 8 months ago

Thanks!!!.

Krishna Jha - 7 years, 8 months ago

area of x^2+y^2 is 100π.sin(x+y)>0 which implies 0>x+y>2π.....like that split the parts in which sin(x+y)>0if you see the graph it form a symetry and if you fold the circle you can see common region is 50π.

Vignesh Subramanian - 7 years, 8 months ago

Yes, even I did think the same way...thanks for ur answer

Krishna Jha - 7 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$