4.Find the area under the curve y = 2x − x^2 from x = 1 to x = 2 using the Midpoint Formula with n = 4.5.Find the area under the curve y = 2x − x^2 from x = 1 to x = 2.6.Evaluate cos x dx.7.Evaluate 2 x dx.8.Evaluate (x^4 − 5x^3 + 3x^2 − 4x − 6) dx.9.Evaluate sin x dx.10.Suppose we are given the following table of values for x and g(x):Use a left-hand Riemann sum with 5 subintervals indicated by the data in the table to approximate g(x) dx.THE MEAN VALUE THEOREM FOR INTEGRALS
As you recall, we did the Mean Value Theorem once before, in Chapter 8, but this time we’ll apply it to
integrals, not derivatives. In fact, some books refer to it as the “Mean Value Theorem for Integrals” or
MVTI. The most important aspect of the MVTI is that it enables you to find the average value of a
function. In fact, the AP Exam will often ask you to find the average value of a function, which is just its
way of testing your knowledge of the MVTI.
Here’s the theorem.