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

1. anonymous

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

2. amistre64

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

3. anonymous

You have to like break it into two peices

4. anonymous

a sector of a circle and a triangle

5. amistre64

oh, then the graph would be useful at anyrate

6. amistre64

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 ....

7. JamesJ

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?

8. anonymous

well the triangle wld be a special triangle

9. JamesJ

No, it would be a right angled triangle.

10. anonymous

1.1 and sqrt(2)

11. anonymous

ya a right triangle

12. JamesJ

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

13. anonymous

|dw:1327361596715:dw|

14. JamesJ

yes, exactly. And its area?

15. anonymous

(b*H)/2

16. anonymous

so it is a 1/2

17. anonymous

ohhhhh I see

18. JamesJ

Nice question.

19. anonymous

Thanks :D