# Definite Integrals: Summation Domination Quiz

*? Put your knowledge to the test. Good luck — the Stickman is counting on you!*

**Definite Integrals***f*(

*x*) = |

*x*| and the

*x*-axis on the interval [-4, 2].

*f*are given in the table below.

Use a left-hand sum with the sub-intervals suggested by the data in the table to estimate the area between the graph of *f* and the *x*-axis on the interval [1, 8].

*f*and the

*x*-axis on [1,10].

*f*(1) +

*f*( 2 ) +

*(3) +*

*f**(4) +*

*f**f*(5) +

*f*(6) +

*(7) +*

*f**(8) +*

*f**(9) +*

*f**f*(10)]

*Δ x*

*f*(1) +

*f*(2)

*(3) +*

*+**f**(4) +*

*f**f*(5) +

*f*(6) +

*(7) +*

*f**(8) +*

*f**(9)]Δ*

*f**x*

*f*(2) +

*f*(3) +

*f*(4) +

*f*(5) +

*f*(6) +

*f*(7) +

*f*(8) +

*f*(9)]Δ

*x*

*f*(2) +

*(3) +*

*f**(4) +*

*f**f*(5) +

*f*(6) +

*(7) +*

*f**(8) +*

*f**(9) +*

*f**f*(10)]Δ

*x*

*f*is increasing on [

*a*,

*b*] then LHS(2) will provide an overestimate for the area between

*f*and the

*x*-axis on [

*a*,

*b*].

*f*is increasing on [

*a*,

*b*] then LHS(2) will provide an underestimate for the area between

*f*and the

*x*-axis on [

*a*,

*b*].

*f*is decreasing on [

*a*,

*b*] then RHS(2) will provide an overestimate for the area between

*f*and the

*x*-axis on [

*a*,

*b*].

*f*is constant on [

*a*,

*b*] then RHS(2) will provide an underestimate for the area between

*f*and the

*x*-axis on [

*a*,

*b*].

*R*be the region between the graph of

*f*and the

*x*-axis on [2, 4]. How many sub-intervals must we use to guarantee that the left-hand sum will be within 0.25 of the exact area of

*R*?

*f*(

*x*) and the

*x*-axis on [

*a*,

*b*]?

*f*(

*x*) = sin

*x*+ 1. Use a trapezoid sum with 4 sub-intervals to estimate the area between

*f*and the

*x*-axis on [0, 2π].

*f*are shown in the table below. We want to estimate the area between

*f*and the

*x*-axis on [0, 4]. We can use the values in the table to find all but one of the following midpoint sums. Which midpoint sum cannot be found using the information in the table?

*f*will any midpoint sum give an overestimate of the area between

*f*and the

*x*-axis on [0, 2]?

*f*(

*x*) =

*x*

^{2}

*f*(

*x*) =

*x*

*f*be a non-negative function that is increasing and concave up. Let

*R*be the region between the graph of

*f*and the

*x*-axis on [

*a*,

*b*]. Which of the following quantities is largest?

*R*.