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AP®︎/College Calculus AB

Course: AP®︎/College Calculus AB > Unit 6

Lesson 6: Applying properties of definite integrals

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Definite integrals properties review

Google Classroom

Review the definite integral properties and use them to solve problems.

What are the definite integral properties?

Sum/Difference:

\int_{a}^{b} [f (x) \pm g (x)] d x = \int_{a}^{b} f (x) d x \pm \int_{a}^{b} g (x) d x

Want to learn more about this property? Check out this video.

Constant multiple:

\int_{a}^{b} k \cdot f (x) d x = k \int_{a}^{b} f (x) d x

Want to learn more about this property? Check out this video.

Reverse interval:

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x

Want to learn more about this property? Check out this video.

Zero-length interval:

\int_{a}^{a} f (x) d x = 0

Want to learn more about this property? Check out this video.

Adding intervals:

\int_{a}^{b} f (x) d x + \int_{b}^{c} f (x) d x = \int_{a}^{c} f (x) d x

Want to learn more about this property? Check out this video.

Practice set 1: Using the properties graphically

Problem 1.1

\int_{- 2}^{0} f (x) d x + \int_{0}^{3} f (x) d x =

units

^{2}

Want to try more problems like this? Check out this exercise.

Practice set 2: Using the properties algebraically

Problem 2.1

\int_{- 1}^{3} f (x) d x = - 2

\int_{- 1}^{3} g (x) d x = 5

\int_{- 1}^{3} (3 f (x) - 2 g (x)) d x =

Want to try more problems like this? Check out this exercise.

Want to join the conversation?

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gcoates0520
Posted 6 years ago. Direct link to gcoates0520's post “How would I graph an equa...”
How would I graph an equation like g(x)=3f(x-2)+1?
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(2 votes)
Answer
- The #1 Pokemon Proponent
  Posted 6 years ago. Direct link to The #1 Pokemon Proponent's post “You need to have some inf...”
  You need to have some info about f(x). I mean how do you graph a function that depends on another function that has not been defined.
  Comment on The #1 Pokemon Proponent's post “You need to have some inf...”
  (7 votes)
lz40247
Posted 3 years ago. Direct link to lz40247's post “If the f(x) inside is f(3...”
If the f(x) inside is f(3x) instead of 3f(x), do you multiply the 3 to the bound values (a and b)? (does then become F(3b)- F(3a)?)
Button navigates to signup pageComment on lz40247's post “If the f(x) inside is f(3...”
(4 votes)
Answer
😊
Posted 5 years ago. Direct link to 😊's post “Proof of. |f(x)g(x)d(x)...”
Proof of.
|f(x)g(x)d(x)=|f(c)g(x)d(x)
c€[a,b].

|~integration from a-b
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(1 vote)
Answer
- Jerry Nilsson
  Posted 5 years ago. Direct link to Jerry Nilsson's post “∫(𝑎, 𝑏) 𝑓(𝑥)𝑑𝑥 and ...”
  ∫(𝑎, 𝑏) 𝑓(𝑥)𝑑𝑥 and ∫(𝑎, 𝑏) 𝑓(𝑐)𝑑𝑥 are not equivalent expressions.
  
  Example:
  𝑎 = 0, 𝑏 = 2
  𝑓(𝑥) = 𝑥 ⇒ ∫(𝑎, 𝑏) 𝑓(𝑥)𝑑𝑥 = ∫(0, 2) 𝑥𝑑𝑥 = 2²∕2 − 0²∕2 = 2
  𝑐 = 2𝑥 ⇒ 𝑓(𝑐) = 2𝑥 ⇒ ∫(𝑎, 𝑏) 𝑓(𝑐)𝑑𝑥 = ∫(0, 2) 2𝑥𝑑𝑥 = 2 ∙ ∫(0, 2) 𝑥𝑑𝑥 = 2 ∙ 2 = 4
  Button navigates to signup page
  (6 votes)
ashwin123robotix
Posted 2 years ago. Direct link to ashwin123robotix's post “how do u integrate xdx be...”
how do u integrate xdx between bounds x and 0
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(1 vote)
Answer
- Venkata
  Posted 9 months ago. Direct link to Venkata's post “It'll use the reverse pow...”
  It'll use the reverse power rule. You integrate xdx to x^(2)/2. Then, substitute the bounds to get 0 - x^(2)/2 = -x^(2)/2
  Button navigates to signup page
  (1 vote)
Z.Ashtab
Posted 6 years ago. Direct link to Z.Ashtab's post “integral f(x) from 3a to ...”
integral f(x) from 3a to 3b equal to integral 3f(3x)from a to b. is this right?
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(0 votes)
Answer
- Pat Florence
  Posted 4 years ago. Direct link to Pat Florence's post “No, this is not true. In...”
  No, this is not true. Integrating from 3a to 3b would mean you are changing the bounds of integration - it totally depends on what the function looks like over the interval from x=3a to x=3b. It may look the same as it does over the interval from x=a to x=b, but odds are it doesn't.
  Button navigates to signup page
  (2 votes)
hongbin.jin66
Posted 5 months ago. Direct link to hongbin.jin66's post “The last property should ...”
The last property should be ∫(𝑎, 𝑏) 𝑓(𝑥)𝑑𝑥 = ∫(𝑎, c) 𝑓(𝑥)𝑑𝑥 + ∫(c, 𝑏) 𝑓(𝑥)𝑑𝑥
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(0 votes)
Answer
- blackberry ❤
  Posted 4 months ago. Direct link to blackberry ❤'s post “Note that they're the sam...”
  Note that they're the same—here b replaces the c in your formula.
  Button navigates to signup page
  (1 vote)
8023834
Posted 3 years ago. Direct link to 8023834's post “$\sqrt{x}$ does this work...”
$\sqrt{x}$ does this work?///
$$ \int_0^8 f(x) = \left[ x + \frac{1}{10} \cdot \frac{x^3}{3} \right]_0^8 = 12 + \frac{12^3}{30} - 0 = 12+56.6 = 69.6 $$
Button navigates to signup pageComment on 8023834's post “$\sqrt{x}$ does this work...”
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Answer