Calculate exact area under curve in Java

To compute it exactly, you would need a computer algebra system library of some sort to perform symbolic manipulations. Such systems are rather complicated to implement, and I am not familiar with any high quality, open source libraries for Java. An alternative, though, assuming it meets your requirements, would be to estimate the area under the curve using the trapezoidal rule. Depending on how accurate you require your result to be, you can vary the size of the subdivisions accordingly.

Numerical integration
There are multiple methods, which can be used. For description, have a look in Numerical Recipes: The Art of Scientific Computing.
For Java there is Apace Commons library, which can be used. Integration routines are in Numerical Analysis section.

Symbolic integration
Check out jScience. Functions module "provides support for fairly simple symbolic math analysis (to solve algebraic equations, integrate, differentiate, calculate expressions, and so on)".
If type of function is given, it can be possible to integrate faster in that specific case than when using some standard library.