Show geometrically why
∫sqrt(2-x^2)
= (pi/4)+(1/2)

Question

Show geometrically why  
∫sqrt(2-x^2)
 = (pi/4)+(1/2)

anonymous · Answer

$$\int\limits_{0}^{1}\sqrt{2-x ^{2}}dx = \Pi /4+1/2$$

amistre64 · Answer

geometrically?  wouldnt that just be a graph of the sqrt part?

anonymous · Answer

You have to like break it into two peices

anonymous · Answer

a sector of a circle and a triangle

amistre64 · Answer

oh, then the graph would be useful at anyrate

amistre64 · Answer

http://www.wolframalpha.com/input/?i=sqrt%282-x%5E2%29%2C+real%2C+x%3D-2to2%2C+y%3D0to2 the wolf hates when I try to define stuff ....

jamesj · Answer

right exactly.  The segment of the circle has angle pi/4, or 1/8 of a complete circle of 2pi and radius sqrt(2).  Hence the segment of circle has area
$$ \frac{1}{8}\pi r^2 = \frac{2}{8}\pi = \frac{\pi}{4} $$

Now, what about the triangle?

anonymous · Answer

well the triangle wld be a special triangle

jamesj · Answer

No, it would be a right angled triangle.

anonymous · Answer

1.1 and sqrt(2)

anonymous · Answer

ya a right triangle

jamesj · Answer

what's the length of the base, b?
And the height, h?

anonymous · Answer

|dw:1327361596715:dw|

jamesj · Answer

yes, exactly.  And its area?

anonymous · Answer

(b*H)/2

anonymous · Answer

so it is a 1/2

anonymous · Answer

ohhhhh I see

jamesj · Answer

Nice question.

anonymous · Answer

Thanks :D