Suppose the mean height of adult women is 67 inches, with a variance of 4 inches. If we know that a woman is at least 72 inches tall, what is her expected height?
My immediate thought was to integrate the PDF of the normal distribution from 72 to infinity. Does that make sense?
Close. If you integrate the normal distribution (with mean of 67 and variance of 4) from $72$ to infinity, you get the total fraction of women over $72$ inches.
You use that as the denominator in your calculation; the numerator is $$ \int_{72}^\infty x N[67;4](x) \, dx $$ where $N[67,4]$ is the probability density and needs to be multiplied by $x$ and integrated, to get an expectation.