Zero net area Consider the function ƒ(𝓍) = 𝓍² ― 4𝓍 .
(a) Graph ƒ on the interval 𝓍 ≥ 0.

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).

(a) Describe the motion of the object over the interval [0,6].

Suppose ƒ is an odd function, ∫₀⁴ ƒ(𝓍) d𝓍 = 3 , and ∫₀⁸ ƒ(𝓍) d𝓍 = 9 .

(a) Evaluate ∫₋₈⁴ ƒ(𝓍) d𝓍 .

Free fall On October 14, 2012, Felix Baumgartner stepped off a balloon capsule at an altitude of almost 39 km above Earth’s surface and began his free fall. His velocity in m/s during the fall is given in the figure. It is claimed that Felix reached the speed of sound 34 seconds into his fall and that he continued to fall at supersonic speed for 30 seconds. (Source: http://www.redbullstratos.com)

(a) Divide the interval [34, 64] into n = 5 subintervals with the gridpoints x₀ = 34 , x₁ = 40 , x₂ = 46 , x₃ = 52 , x₄ = 58 , and x₅ = 64. Use left and right Riemann sums to estimate how far Felix fell while traveling at supersonic speed.

The velocity in ft/s of an object moving along a line is given by v = ƒ(t) on the interval 0 ≤ t ≤ 6 (see figure), where t is measured in seconds.

(a) Divide the interval [0,6] into n = 3 subintervals, [0,2] , [2,4] and [4,6]. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the right endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,6] (see part (a) of the figure)                                                                                                             

Sigma notation Evaluate the following expressions.

(a)    10                                                                                                                                                                               

       ∑ κ                                                                                                                                                                          

       κ=1                         

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(a) If ƒ is a constant function on the interval [a,b], then the right and left Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n.