The Fundamental Theorem of Calculus An Introduction to A Stand-Alone Instructional Resource (STAIR) review by Grace Zhang and Sarah Oesterling

Lesson Objectives In this review, you will… Learn the different parts of the Fundamental Theorem of Calculus, also known as FTC Discover how to apply the FTC in both calculator and non-calculator problems Better understand the meaning of the problems through FTC, particularly in word problems

Pre-Quiz! Prerequisites Students must have a basic understanding of the common antiderivative rules, along with how to solve indefinite integrals. Test your knowledge! Pre-Quiz!

Common Rules Quiz If 𝑓 𝑥 = 𝑥 𝑛 Then what is the anti-derivative of 𝑓(𝑥)? a) 𝑥 𝑛+1 𝑛+1 b) 𝑥 𝑛−1 𝑛+1 c) 𝑥 𝑛−1 d) 𝑥 𝑛+1 𝑛−1

The 𝑛 represents any real number Correct! 𝑓 𝑥 𝑑𝑥= 𝑥 𝑛+1 𝑛+1 a) The 𝑛 represents any real number Example: 3 𝑥 2 = 3𝑥 3 3 = 𝑥 3

Incorrect! …. Try again! Hint 1: The opposite step to a derivative 𝑓 𝑥 𝑑𝑥= 𝑥 𝑛+1 𝑛+1 Hint 3: Incorrect! …. Try again!

Common Rules Quiz 2. If 𝑓 𝑥 = 𝑒 𝑥 , what is the antiderivative of 𝑓 𝑥 ? a) 𝑒 𝑥+1 𝑥+1 b) 𝑒 𝑥 c)𝑥 𝑒 𝑥 d)𝑒

Correct! b) The antiderivative of 𝑓 𝑥 = 𝑒 𝑥 equals 𝑒 𝑥 𝑑𝑥 which equals 𝑒 𝑥 The antiderivative of 𝑒 𝑥 is always just as it is: 𝑒 𝑥

2. If 𝑓 𝑥 = 𝑒 𝑥 , what is the antiderivative of 𝑓 𝑥 ? Hint 1: 𝑒 𝑥 uses different properties, and e can not be treated as an x or a number value. What is the different rule that follows e? Hint 2: Incorrect! …. Try again!

Common Rules Quiz 3. What is the antiderivative for –cos(x) ? a) cos(x) c) tan(x) b) sin(x) d) –sin(x)

A helpful way to remember this rule is to draw this out. Correct! d) −𝑐𝑜 𝑠 𝑥 𝑑𝑥= -sin (x) S C -S -C A helpful way to remember this rule is to draw this out.

Incorrect! …. Try again! 3. The antiderivative of –cos(x) = Sin Cos -Sin -Cos Incorrect! …. Try again!

Common Rules Quiz 4. 𝑠𝑒𝑐 2 𝑥 𝑑𝑥 = ? a) tan(x) b) 𝑐𝑠𝑐 2 4. 𝑠𝑒𝑐 2 𝑥 𝑑𝑥 = ? a) tan(x) b) 𝑐𝑠𝑐 2 c) sec 𝑥 tan⁡(𝑥) d) csc 𝑥 cot⁡(𝑥)

Correct! 𝑠𝑒𝑐 2 𝑥 𝑑𝑥= tan(x) For these rules, the best way to apply them is to simply memorize them. * The “+ C” shown in the chart is included to give a result of the actual number, rather than the number between a specific integral.

Incorrect! …. Try again! 𝑠𝑒𝑐 2 𝑥 𝑑𝑥= sec⁡(𝑥) 2 𝑑𝑥 4. For these rules, the best way to apply them is to simply memorize them. * The “+ C” shown in the chart is included to give a result of the actual number, rather than the number between a specific integral. Incorrect! …. Try again!

End of Quiz Great Job! To review… 𝑥 𝑛 𝑑𝑥= 𝑥 𝑛+1 (𝑛+1) 𝑒 𝑥 𝑑𝑥= 𝑒 𝑥 Sin Cos -Sin -Cos

Fundamental Theorem of Calculus Part II The antiderivative (integral) is used to measure the area under a function with specific intervals Since 𝑓(𝑥) =𝐹 𝑥 , then 𝑎 𝑏 𝑓(𝑥) =𝐹 𝑏 −𝐹(𝑎)

Explanations continued… The area under the curve for a line might seem unnecessary for an integral, because it can be found with a simple area formula (top right) However, when the graphs become more complicated, dividing a section up into exact geometric parts are almost impossible, so integral is the best and most simple to find a value under the curve. (bottom right)

Indefinite vs. Definite Indefinite Integrals Definite Integrals Results in another equation Variable representations Involves only one step with integral rule Results in a real number/ value 2 steps involving integral rule and basic algebra

Practice! Evaluate 0 3 3 𝑡 2 dt a) 27 c) 9 b) 30 d) 18

CORRECT! 

Hint 1: convert 3 𝑡 2 to its indefinite integral Incorrect  Hint 1: convert 3 𝑡 2 to its indefinite integral Hint 2: Top- Bottom Hint 3: 3 𝑡 3 3 = 𝑡 3

*RULE If 𝑎 𝑏 𝑓 𝑡 𝑑𝑡 is given and 𝑏<𝑎 add negative in front of the equation and reserve the top and bottom numbers so it’s − 𝑏 𝑎 𝑓 𝑡 𝑑𝑡 The reason for the switch has to do with the conceptual part of finding the area under the curve with a graph. Finding an integral backwards, for example, on intervals [3,0] doesn’t make sense, so the order of the interval has to be switched to [0,3]. The magnitude of the answer doesn’t change, but the sign is now opposite.

Evaluate 3 −1 ( 3𝑥 2 −2𝑥+5) 𝑑𝑥 a) –16 b) -40 c) -39 d) -29

Correct! = [ 𝑥 3 − 𝑥 2 +5𝑥] 3 −1 b) 3 −1 ( 3𝑥 2 −2𝑥+5) 𝑑𝑥 = -40 = [ 𝑥 3 − 𝑥 2 +5𝑥] 3 −1 = (-1-1-5) - (27-9+15) = -6-33 = -40

Hint 2: Plug in -1 and -3 into x. Incorrect  Hint 1: First take the antiderivative of the function. Hint 2: Plug in -1 and -3 into x. Hint 3: Remember, 𝑠 𝑝 𝑓′(𝑥)𝑑𝑥=𝑓 𝑝 −𝑓(𝑠)

FTC 1 You learned that… 𝑎 𝑏 𝑓 𝑥 𝑑𝑥=𝐹 𝑎 −𝐹 𝑏 =𝐹(𝑥) But what if you want to take the Derivative of a definite integral, like shown below? If ℎ 𝑥 = 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 Then ℎ ′ 𝑥 = ?

ℎ 𝑥 = 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 ℎ ′ 𝑥 =𝑓(𝑥) FTC 1 If f is continuous on [a, b] h is continuous on [a, b] and differentiable on (a, b) ℎ 𝑥 = 𝑎 𝑏 𝑓 𝑥 𝑑𝑥 ℎ ′ 𝑥 =𝑓(𝑥)

FTC 1 The FTC rule can be understood as substitution because the action can be simplified to directly switching one variable for another. Hence However, if the variable contains a function within, the derivative must be taken and multiplied on to the substitution. 𝑎 𝑥 𝑓 𝑡 𝑑𝑡=𝑓(𝑥)

https://www.ibiblio.org/kupha ldt/electricCircuits/Ref/REF_6. html http://calculus.nipissingu.ca/tu torials/integrals.html https://www.wyzant.com/reso urces/lessons/math/calculus/in tegration/finding_volume