Graph the function $f(x)=−2x^2+8x$. . 

1. Approximate the area under the curve for $0\le x\le4$ using four trapezoids.

   Hint (a):

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

   Hint (a):

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

   Hint (a):

   The bases are determined by the function.

   Hint (a):

   $\frac{1}{2}(1)[f(0)+2(1)+2f(2) + 2f(3) + f(4)]$

2. Write a Riemann sum to approximate the area under the curve for $0\le x\le4$ using four left endpoint rectangles. Then, use the summation feature of your graphing calculator to evaluate the sum. 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.