4.9-5.3 Study Guide for Quiz
Name__________________________________Date____________________Section_______
Part I: Antiderivatives and Riemann Sum
1. Estimate the area under the graph of 𝑓(𝑥) = 𝑥 2 − 2𝑥 + 1from x = 1 to x = 5 using four
approximating rectangles and right endpoints, left endpoints and the midpoint rule.
Sketch the graph and the rectangles. Is your estimates an underestimate or an
overestimate?
2.
If 𝑓(𝑥) = 𝑠𝑖𝑛(5𝑥), 0 ≤ 𝑥 ≤ 𝜋 evaluate the left and right hand Riemann sum with n = 6.
3. The acceleration of a body due to gravity is constant and can be described by 𝑎(𝑡) =
−9.8 Find the position of a ball as a function of time if you threw the ball straight
𝑚
upwards at a velocity of 50 𝑠 ( 𝑣(0) = 50 ) and you are standing on building that is 20
meters high ( 𝑝(0) = 20 ).
4. Translate the definite integral into a Riemann Sum.
3
a) ∫1 √4 + 𝑥 2 𝑑𝑥
𝜋
b) ∫0 sin(𝑥) 𝑑𝑥
Part II: Definite Integrals
Compute the following definite integrals:
3
1. ∫1 (𝑥 2 + 2𝑥 − 4) 𝑑𝑥
2.
1
∫−1 𝑥100 𝑑𝑥
𝜋
2. ∫03 sec(𝑥)tan(𝑥) 𝑑𝑥
4.
3
∫0 2𝑠𝑖𝑛𝑥 − 𝑒 𝑥 𝑑𝑥
Give the middle approximation with 3 subintervals for 𝑓 on the interval [0, 20]