Estimate $A(f(x)$, $-3 ≤ x ≤ 3$) for $f(x) = 2x^2 + 1$. .

1. Using left endpoint rectangles. The first two rectangles are shown.

   Hint 1(a):

   Regarding the word ESTIMATE: You are not being asked to make a guess. You still need to compute. Of course, the area using rectangles is an approximation of the actual area under $f(x)$.

   Hint 2(a):

   Sketch the remaining rectangle. Notice: The height of the LAST rectangle is evaluated at $x = 2$ (not $x=3$).

   Hint 3(a):

   Each rectangle has a base of $1$ and a height of $f(x)$. Area $= 1(f(-3) + f(-2) + f(-1) + f(0) + f(1) + f(2)) =$_____________

2. Using right endpoint rectangles.

   Hint (b):

   Notice: The height of the FIRST rectangle is evaluated at $x = -2$. The height of the LAST rectangle is evaluated at $x = 3$.

   More Help (b):

   Each rectangle has a base of $1$ and a height of $f(x)$. Area $= 1( f(-2) + f(-1) + f(0) + f(1) + f(2) + f(3)) =$_____________

3. Using trapezoids. What do you notice? Does this always happen?

   Hint (c):

   More Help (c):

   The trapezoidal sum is the AVERAGE of the left- and right-endpoint sums! Expain why this happens (algebraically and geometrically).