Scribd
Upload
14 views4 pages

Calculus: Riemann Sums & Integrals

The document discusses the concept of approximating the area under a curve using Riemann sums and summation notation in calculus. It explains how to represent the area under the curve of a function using limits and provides examples of rewriting definite integrals in summation notation and vice versa. Additionally, it includes practice problems related to definite integrals and Riemann sums.

Uploaded by

ah01211393916
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
14 views4 pages

Calculus: Riemann Sums & Integrals

The document discusses the concept of approximating the area under a curve using Riemann sums and summation notation in calculus. It explains how to represent the area under the curve of a function using limits and provides examples of rewriting definite integrals in summation notation and vice versa. Additionally, it includes practice problems related to definite integrals and Riemann sums.

Uploaded by

ah01211393916
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 4

Calculus 6.

3 Summation Notation Notes


Write your questions
and thoughts here!
The graph of the function 𝑓 𝑥 is shown below. How can we approximate the area under the
curve on the interval 1, 6 ?

Create a Reimann sum with 𝑛 subintervals



and have 𝑛 → ∞.





        



If 𝑛 is the number of subintervals If 𝑛 → ∞ on the interval 𝑎, 𝑏 , what does


(rectangles) on the interval 𝑎, 𝑏 , what is the width of each subinterval (rectangle)
the width of each subinterval (rectangle)? approach?

𝑏 𝑎 ∆𝑥 → 0
∆𝑥
𝑛

The sum of the area of all rectangles gives you the area under the curve.

∆𝑥 ⋅ 𝑓 𝑥 ∆𝑥 ⋅ 𝑓 𝑥 ∆𝑥 ⋅ 𝑓 𝑥 ⋯ ∆𝑥 ⋅ 𝑓 𝑥

We can represent this area by combining limits with summation notation.

Summation Notation

The area under the curve of 𝑓 𝑥 on the interval 𝑎, 𝑏 is represented by


𝑏 𝑎 𝑏 𝑎
lim ⋅𝑓 𝑎 𝑘
→ 𝑛 𝑛

where 𝑛 represents the number of subintervals (rectangles) there are in the interval 𝑎, 𝑏 and
𝑘 represents the 𝑘th subinterval.

Another way of writing the summation notation

lim ∆𝑥 ⋅ 𝑓 𝑥
∆ →
Write your questions
and thoughts here!
Definite Integral Notation
𝑏 𝑎 𝑏 𝑎
lim ⋅𝑓 𝑎 𝑘 𝑓 𝑥 𝑑𝑥
→ 𝑛 𝑛
The area under the curve of 𝑓 𝑥 on the interval 𝑎, 𝑏 is represented by 𝑓 𝑥 𝑑𝑥.

Examples:
1. Rewrite the definite integral using summation notation.
4 4
𝑥 3 𝑑𝑥 lim 2 𝑘 3
→ 𝑛 𝑛

2. Rewrite the summation notation expression as a definite integral.


6 6𝑘
lim 4
→ 𝑛 𝑛

a. b. c.
4 𝑥 𝑑𝑥 𝑥 𝑑𝑥 6 4 6𝑥 𝑑𝑥

3. lim ⋯
→
Assuming the lower limit “a” is 0, write a definite integral that represents the above
expression.
1
𝑑𝑥
𝑥 3

4. The expression cos cos cos ⋯ cos is a Riemann sum


approximation for what definite integral?

cos 𝑥 𝑑𝑥

Where is the 10? Why isn’t it written in the integral?


6.3 Summation Notation
Calculus
Practice
Write a definite integral that is equivalent to the given summation notation. The lower limit for the
integral is also given to help you get started.
1. Integral’s lower limit 0 2. Integral’s lower limit 3 3. Integral’s lower limit 6
𝜋 𝜋 5 5𝑘
lim cos 𝑘 lim 3 9 1
→ 𝑛 𝑛 → 𝑛 𝑛 lim
→ 𝑛 9𝑘
6
𝑛

Write a summation notation equivalent to the definite integral.


4. 𝑥 𝑑𝑥 5. 𝑑𝑥 6. √𝑥 𝑑𝑥

7. Which of the following expressions is equal to lim 𝑒 𝑒 𝑒 ⋯ 𝑒 ?


→

(A) 𝑒 𝑑𝑥 (B) 𝑒 𝑑𝑥

(C) 𝑒 𝑑𝑥 (D) 𝑒 𝑑𝑥

8. The expression sin sin sin ⋯ sin is a Riemann sum approximation of


which of the following integrals?

(A) 𝑥 sin 𝑥 𝑑𝑥 (B) 𝑥 sin 𝑥 𝑑𝑥

(C) sin 𝑥 𝑑𝑥 (D) sin 𝑥 𝑑𝑥


9. The expression ln 2 ln 2 ln 2 ln 2 ln 2 is a Riemann sum
approximation of which of the following integrals?

(A) ln 𝑑𝑥

(B) ln 𝑥 𝑑𝑥

(C) ln 𝑥 𝑑𝑥

(D) ln 𝑥 𝑑𝑥

(E) ln 2 𝑥 𝑑𝑥

6.3 Summation Notation Test Prep


4𝑘 4
10. Which of the following definite integrals are equal to lim
→
1
𝑛 𝑛

I. 𝑥 𝑑𝑥

II. 1 𝑥 𝑑𝑥

III. 4 1 4𝑥 𝑑𝑥

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) I, II, and III only

You might also like