I have a short writeup on using integration to find the area of a circle (see attached file). In the third method (integrating sqrt(R^2 - x^2)), I have to switch the limits of integration to get a positive area. How do I justify this?

Question

I have a short writeup on using integration to find the area of a circle (see attached file).  In the third method (integrating sqrt(R^2 - x^2)), I have to switch the limits of integration to get a positive area.  How do I justify this?

anonymous · Answer

attachment

amistre64 · Answer

if f(a) is greater than f(b) then ...  f(b)-f(a) < 0

experimentx · Answer

?? which line??

amistre64 · Answer

$$\int_{0}^{1}dx=1-0$$
$$\int_{1}^{0}dx=0-1$$

amistre64 · Answer

since area itself is an absolute value, just wrap it in a modulus :)

experimentx · Answer

my favouraite http://mathworld.wolfram.com/images/eps-gif/CircleAreaStrips_1000.gif

anonymous · Answer

But the original integral goes from x=0 to x=R, F( R) > 0
while after substitution, it goes from theta = pi/2 to theta = 0
so, I just say "guess we have to switch signs here"  ?
is there a deeper rationale?

anonymous · Answer

@experimentX thanks for that, had forgotten it  :)

experimentx · Answer

|dw:1344780739855:dw|
the coordinates have been changed from cartesian to polar.