Question

用定积分计算圆的面积如题,但是√(r^2-x^2)的原函数是什么呢？

Answer 1

以x^2+y^2=r^2为例:

4∫[0～r]√(r^2-x^2)dx

上式可用换元法发来算,我以为你会呢,所以没写,!

设:x=rsint

则上式变为4∫[0～π/2]rcostd(rsint)

=4∫[0～π/2]r^2(cost)^2dt

=4r^2∫[0～π/2](1+cos2t)/2dt

=4r^2(π/4+∫[0～π/2](cos2t/4)d(2t))

=4r^2*π/4

=πr^2