Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.
(b) Find the mass of the right half of the rod (5 ≤ x ≤ 10) .

Area functions for the same linear function Let ƒ(t) = t and consider the two area functions A(𝓍) = ∫₀ˣ ƒ(t) dt and F(𝓍) = ∫₂ˣ ƒ(t) dt .

(b) Evaluate F(4) and F(6). Then use geometry to find an expression for F (𝓍) , for 𝓍 ≥ 2. 

Sigma notation Evaluate the following expressions.                                                                                                                                          

(b)    10                                                                                                                                                                               

       ∑  (2κ + 1)                                                                                                                                                                          

       κ=1                         

Displacement from a table of velocities The velocities (in mi/hr) of an automobile moving along a straight highway over a two-hour period are given in the following table.

(b) Find the midpoint Riemann sum approximation to the displacement on [0,2] with n = 2 and .n = 4 .

Area functions for linear functions Consider the following functions ƒ and real numbers a (see figure).

b) Verify that A'(𝓍) = ƒ(𝓍).

ƒ(t) = 4t + 2 , a = 0

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)

(b) 4 + 5 + 6 + 7 + 8 + 9

{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.

f(𝓍) = x³ on [-1,2]

(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.