Sketch the function $f(x) =-2x^2 + 8x$.

1. Estimate $A(f, 0 ≤ x ≤ 4)$ using four trapezoids.

   Hint 1(a):

   $\text{Area of a trapezoid }=\frac{1}{2}h(b_1+b_2).$

   Hint 2(a):

   On the given domain, what will be the height of each trapezoid?

   Hint 3(a):

   The bases are determined by the function.

2. Write a Riemann sum representing $A(f, 0 ≤ x ≤ 4)$ using $4$ left endpoint rectangles. Then, use the summation feature of your graphing calculator to evaluate. Compare the accuracy of the trapezoids and the rectangles.

   Hint (b):

   $1[f(0) + f(1) + f(2) + f(3)]$

3. Will the approximation with trapezoids always equal the approximation with rectangles for all functions? Why or why not?

   Hint (c):

   Think about the symmetry.