[Notes & HW Answers] 5.4 Theorems about Definite Integral

[Notes & HW Answers] 5.4 Theorems about Definite Integral.pdf

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[Prepwork 5.4]

Q1. For some function f, suppose that for some a<b<c, ∫^b_a f(x) dx=7 and ∫^c_a f(x) dx=9. Find each of the following.

A: ∫^a_b f(x) dx= −7 ;∫^c_b f(x) dx= 2  

 

Q2. Use the technique of example 2 in the text to evaluate the integral ∫^2_(−2) (|2x|+3) dx exactly.

 

A: ∫^2_−2 (|2x|+3) dx = 20

 

 

Q3. Using a calculator to evaluate the appropriate integral, find the average value of P = f(t) = 2.03(1.05)^t for 0≤t≤25.

 

A: Average value of P= 3.97153862

 

 

[HW 5.4]

Q1. Let ∫^b_a f(x) dx =18 and ∫^b_a (f(x))^2 dx = 4. Find the following integrals:
∫^b_a cf(z) dz= 18c
∫^b_a (f(x))^2 dx − (∫^b_a f(x) dx)^2= −320
∫^(b+3)_(a+3) f(x−3) dx=18

Q2. Find the average value of f(x)=2x+3 over [5,10]

A: average value = 18

 

Q3. Find the area of the region between y=x^(1/2) and y=x^(1/3) for 0≤x≤1.

A: area = (3/4)​(1)^(4/3​)−(3/4)​(0)^(4/3​)−(2/3)​(1)^(3/2​)+(2/3)​(0)^(3/2​)

 

Q4.  Let ∫^5_0 f(x) dx = 7.

(a) What is the average value of f(x) on the interval from x=0 to x=5?
average value = 7/5​

(b) If f(x) is even, find each of the following:
∫^5_(−5) f(x) dx= 14
the average of f(x) on the interval x=−5 to x=5 = 7/5​

(c) If f(x) is odd, find each of the following:
∫^5_(−5) f(x) dx= 0
the average of f(x) on the interval x=−5 to x=5= 0

 

Q5. Use the figure below, which shows the graph of 𝑦=𝑓(𝑥), to answer the following questions.

A. Estimate the integral: ∫^3_(−3) 𝑓(𝑥) 𝑑𝑥≈ −3.6

B. Which of the following average values of 𝑓f is larger?
(1) Between 𝑥=−3 and 𝑥=3
(2) Between 𝑥=0 and 𝑥=3 

 

Q6. The figure below to the left is a graph of 𝑓(𝑥), and below to the right is 𝑔(𝑥).

𝑓(𝑥)	𝑔(𝑥)

(a) What is the average value of 𝑓(𝑥) on 0≤x≤2?
avg value = 1/4​

(b) What is the average value of 𝑔(𝑥) on 0≤x≤2?
avg value = 3/8​

(c) What is the average value of 𝑓(𝑥)⋅𝑔(𝑥) on 0≤x≤2?
avg value = 0

(d) Is the following statement true?

Average(𝑓)⋅Average(g)=Average(𝑓⋅𝑔)

A. Yes
B. No

Q7. If 𝑓(𝑥) is odd and ∫^6_(−2) 𝑓(𝑥) 𝑑𝑥= 14, then

A: ∫^6_2𝑓(𝑥) 𝑑𝑥= 14

Q8: If ∫^5_3 (8𝑓(𝑥)+5) 𝑑𝑥=14, then
A: ∫^5_3 𝑓(𝑥) 𝑑𝑥= 1/2