Y2-y1 divides by x2-x1 = 4/9
To evaluate the integral, rewrite the integrand as
Recall that
The leftmost sum is the well-known power series expansion for the function . In the rightmost sum, we just replace with .
This particular power series has a property called "uniform convergence". Roughly speaking, it's a property that says a sequence of functions converges to some limiting function in the sense that and get arbitrarily close to one another. If you have an idea of what "convergence" alone means, then you can think of "uniform convergence" as a more powerful form of convergence.
Long story short, this property allows us to interchange the order of summation/integration to write
The integral can be tackled with a substitution,
so that the integral is equivalent to
The remaining integral reduces to , which you can derive for yourself via integration by parts/power reduction.
So we have
which is the same as
and hence the identity.
It is C. 498.83 units cubed.
Hope I helped? ^ω^