### Home > CALC > Chapter 1 > Lesson 1.2.3 > Problem 1-58

Estimate

*A*(*f*(*x*), −3 ≤*x*≤ 3) for*f*(*x*) = 2*x*^{2}+ 1. 1-58 HW eTool (Desmos). Homework Help ✎

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

Using right endpoint rectangles.

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

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*).

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

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)) =_____________

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

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)) =_____________

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

Use the eTool below to view the graphs.

Click the link at right for the full version of the eTool: Calc 1-58 HW eTool