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16 Sep 2014, 01:16
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What is the area of the region enclosed by lines \(y=x\), \(x=-y\), and the upper crescent of the circle \(y^2+x^2=4\) ?
A. \(\frac{\pi}{4}\)
B. \(\frac{\pi}{2}\)
C. \(\frac{3\pi}{4}\)
D. \(\pi\)
E. \(4\pi\)
A. \(\frac{\pi}{4}\)
B. \(\frac{\pi}{2}\)
C. \(\frac{3\pi}{4}\)
D. \(\pi\)
E. \(4\pi\)
16 Sep 2014, 01:16
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Bookmarks
Official Solution:
What is the area of the region enclosed by lines \(y=x\), \(x=-y\), and the upper crescent of the circle \(y^2+x^2=4\) ?
A. \(\frac{\pi}{4}\)
B. \(\frac{\pi}{2}\)
C. \(\frac{3\pi}{4}\)
D. \(\pi\)
E. \(4\pi\)
The circle represented by the equation \(x^2+y^2 = 4\) is centered at the origin and has the radius of \(r=\sqrt{4}=2\).
Look at the diagram below:
We need to find the area of the upper crescent, so the area of the yellow sector. Since the central angle of this sector is 90 degrees then its area would be 1/4 of that of the circle (since circle is 360 degrees).
The area of the circle is \({\pi}{r^2}=4\pi\), 1/4 of this value is \(\pi\).
Answer: D
What is the area of the region enclosed by lines \(y=x\), \(x=-y\), and the upper crescent of the circle \(y^2+x^2=4\) ?
A. \(\frac{\pi}{4}\)
B. \(\frac{\pi}{2}\)
C. \(\frac{3\pi}{4}\)
D. \(\pi\)
E. \(4\pi\)
The circle represented by the equation \(x^2+y^2 = 4\) is centered at the origin and has the radius of \(r=\sqrt{4}=2\).
Look at the diagram below:
We need to find the area of the upper crescent, so the area of the yellow sector. Since the central angle of this sector is 90 degrees then its area would be 1/4 of that of the circle (since circle is 360 degrees).
The area of the circle is \({\pi}{r^2}=4\pi\), 1/4 of this value is \(\pi\).
Answer: D
