standard name for area under type 1 vs type 2 error curve? we already have precision-recall, and roc curves, why aren't type 1-type 2 error curves more popular? 
Is there a standard name for the area under such plots? 
 A: You ask two questions here. Regarding its name: I don't know, whether there is some established name for it, but something as type 1-type 2 error curve or type-I/II-error trade-off might describe it well enough.
I will focus on why such plots aren't as popular as ROC curves or popular at all.
There at least two major problems involved. First, one simply does not know the type II error because it depends on the exact truth, which is unknown. This is an important difference to ROC-curves, where one knows the truth. Second, to increase the type I error rate one needs to increase the threshold for the p-value to be considered as "significant". This is a severe violation of the general principle that the significance level must be set before calculating any p-values to guarantee the type I error rate.
There is nothing wrong in using such type-I/II-error trade-off to determine a suitable sample size in the planning phase of the study. Personally, I have never seen this because fixed numbers as 5% type I error rate and 80% power have been established. People rather look at variations of the alternative hypothesis and sample size than at variation of type I/II error.
Actually, every software that I know off which comes with a sample size calculation tool has some option to plot the power as a function of sample size or effect size but rarely as a function of the type I error rate. Just as an example, this is a SAS plot showing the power under variations of the alternative hypothesis for fixed type I error rate and sample size:

https://blogs.sas.com/content/iml/2013/06/05/simulation-power-curve.html#prettyPhoto
Some further examples with either fixed type I error or fixed type II eror can be found here: How to best display graphically type II (beta) error, power and sample size?
