How to calculate the area under a Weibull curve

Hi all,

I'm practising my survival analysis skills and I'm trying to conduct a partitioned survival model by means of a Weibull distribution. I can easily manage to do so for exponential distributions, where the survival function is denoted as s(t)=EXP(-(EXP(-intercept)*t)). The area under the curve is then A(t)=(1-s(t))/EXP(-intercept). The intercept is derived from an analysis in R.

The survival curve for a Weibull distribution is S(t)=EXP(-EXP(-(intercept/EXP(log-scale)))*t^(1/EXP(log-scale))).
The intercept and log-scale are again derived from R. I do not have a formula to calculate an area under this survival curve, nor can I find one. Could anyone help me with this?

Thanks!

*I've posted the same question on the mrexel forum. I can't post the link since I don't have a sufficient amount of posts yet.