More definite Integrals and Area under a Curve (4.4 p 293 # 62)

 More Definite Integrals and the Area Under a Curve (4.4 p 293 # 62)

With just a little information about the area of some regions, we can use the properties of integrals to figure out 6 different things!

MVT for Integrals, 1st FTC, and 2nd FTC Proofs

MVT for Integrals, 1st FTC, and 2nd FTC Proofs 

Section 4.4 is chock full of gold.  There is a lot there, so the video lets you take it in.

Here are the Subjects by Time:

 0:00 - The MVT for integrals (Average Value Thm)  & AVERAGE VALUEs

 4:20 - An example with f(x) =6 and a curious observation

 6:24 -The First Fundamental Theorem Proof

11:16 - An Example of FTC1 

12:51 - "Net Change Theorem" version of FTC1 with a "word problem" example

15:27 -The Second Fundamental Theorem of Calculus with for examples

19:05 - The End

U-substitution with a Definite Integral (4-R p310 #67)

 U-substitution with a Definite Integral (4-R p310 #67)

When you use u-substitution with a definite integral, you don't have to switch back to make an expression in terms of x.  Since any definite integral is the signed area, it is just a number.  After finding the anti-derivative in terms of you, the FTC can be used to arrive at the answer. 

Using the u-substitution Technique with an Indefinite Integral (4-R p301 #59)

 Using the u-substitution Technique with an Indefinite Integral (4-R p301 #59)

A basic example of how to integrate something that looks like a product, where one factor has something in common with the derivative of the other factor.

Basic Use of the First Fundamental Theorem (4-R p 310 #41)

 Basic Use of the First Fundamental Theorem (4-R p 310 #41)

Here we use the First Fundamental Theorem of Calculus to evaluate a definite integral, and we will check our work with a TI-84 calculator,

Riemann Sums to Estimate an Integral (4-R p 309 #25)

 Riemann Sums to Estimate an Integral (4-R p 309 #25)

Here we use left and right Riemann sums to get the upeer and lower bounds on an integral. We can then use the TI-84 to check our work.

Mean Value Theorem for Integrals with the TI-84 (4-5#85)

Mean Value Theorem for Integrals with the TI-84 (4-5#85)

Estimating Average Sales with t he TI-84 and the Mean Value Theorem for Integrals (4-5#85)

A u-Substitution Using the Same U Formula Two Ways (4.5 p. 306 #57)

 A u-Substitution Using the Same U Formula Two Ways (4.5 p. 306 #57)

A Definite Integral with u-Substitution (4-5 p 305 #61)

A Definite Integral with u-Substitution (4-5 p 305 #61).  

This one has a surprising result!

Some Complicated Integrals That Are Easier than they Appear! (4-5 p 307 #81)

 4-5 p 307 #81  Some Complicated Integrals That Are Easier than they Appear!

4-5 p 307 #81  Some Complicated Integrals That Are Easier than they Appear!  U-substitution to the rescue!

Using the Chain Rule with an Accumulator Function (4-4 p294 #83, 85)

Using the Chain Rule with an Accumulator Function (4-4 p294 #83, 85)

 If the derivative of the upper limit has a derivative more complicated than 1, then your better use the chain rule!

An Accumulaotr function without a constant 4-4 (p. 294 #81)

 

An Accumulaotr function without a constant 4-4 (p. 294 #81)

Without one of the limits being a constant, you can't use the 2nd FTC... what to do?  Invent one!

An Accumulation function (4.4 p. 293# 67)

An Accumulation function (4.4 p. 293# 67)

 An Accumulation function is a function that is a function of the accumulation of area under a curve.  A nice feature is that the derivative is based on the integrand itself

Estimating a Definite Integral (4.3 p278 #53)

 

Estimating a Definite Integral (4.3 p278 #53)

Estimating a definite integral with the area of rectangles. 

Integrals as Area Above and Below the x-axis (4.3 p. 278 #47)

 Integrals as Area Above and Below the x-axis (4.3 p. 278 #47)

Area can be negative now.... either by being below the x-axis, or by going backwards with area above the x-axis.  

Properties of Integrals (4.3 p 279 #49)

 Properties of Integrals (4.3 p 279 #49)

Here we demonstrate breaking apart integrals, changing the limits and capitalizing the properties of odd and even functions.

Review of Sigma Sums (4.2 p 267 #9, 22)

 Review of Sigma Sums (4.2 p 267 #9, 22)

We explain some useful sum formulas, work some examples, then show how to use the TI-84 to confirm our result.

A Particle Moves along the x-Axis (4.1 p. 256 #65)

 A Particle Moves along the x-Axis (4.1 p. 256 #65)

We often have this one-dimensional particle that moves along the x-axis as an AP exam question, and a number of things can be surmised fromthe first and second derivatives....

How High Will It Go (in Europe) (4.1 p. 256) # 60

 

How High Will It Go (in Europe) (4.1 p. 256) # 60

This time we find the maximum height of a ball thrown into the air in meters.  All we need to know is the velocity and height when it is thrown! In this one we use the constant acceleration of 9.8 meter per second per second.

How High will It Go? (4.1 (p 256 #57)

 How High will It Go? (4.1 (p 256 #57)

We find the maximum height of a ball thrown into the air.  All we need to know is the velocity and height when it is thrown! In this one we use the constant acceleration of 32 feet per second per second.