Is the approximation and underestimate or an overestimate?

Question

A left Riemann sum with n=6 subintervals is used to approximate the area under the curve of f(x)=−4x2+64x+3 over the interval [6,7]. Is the approximation an underestimate or an overestimate?

Answer 1

Check the first derivative. It is positive throughout the interval as is the function. The left side of every rectangle will be the minimum for the range (7-6)/6 for each rectangle, leading to underestimating the area.