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

Related Rated Samples

 Related Rates Example Problems

Here are 7 examples of  Related Rates problems.

Using L'Hôpital's Rule on an e^x Function

Using L'Hôpital's Rule on an e^x Function

 

Inverse Functions Have Reciprocal Slopes (5.R p 400 #39)

 Inverse Functions Have Reciprocal Slopes (5.R p 400 #39)

A u-sub for arctan (5-R p. 400 #109)

 A u-sub for arctan (5-R p. 400 #109)

A Hybrid Parametric Area with d(theta) (10-R p.747 #57)

 A Hybrid Parametric Area with d(theta) (10-R p.747 #57)

Elliptical Orbit in a Polar Form (10.6 p745 #59)

 

Elliptical Orbit in a Polar Form (10.6 p745 #59)  

Not only do we answer the question at hand, we derive the polar form of a conic section.

If you want to skip ahead: 

Minute 5:45 Finding the distance between the surface of Earth and Explorer 18 when the angle is 60 degrees

Minute 12:53 Proof that e = c/a = 2c/2a=(distance between focii/major axis) 

Definite Integral with arcsine (5.8 p387 #33)

 Definite Integral with arcsine (5.8 p387 #33)

Finding a Tangent Line Implicitly on a function with e^x (5.4 p348 #65)

 Finding a Tangent Line Implicitly on a function with e^x (5.4 p348 #65)

Implicit Differentiation with e^x (5.4 p 348 #63)

 Implicit Differentiation with e^x (5.4 p 348 #63)

Verifying a Differential Equation involoving e^x (5.4 p349 #69)

 Verifying a Differential Equation involoving e^x (5.4 p349 #69)

A Bounded Area (def Int) with log base 4 (5.5 p359 #81)

 A Bounded Area (def Int) with log base 4 (5.5 p359 #81)

This one has u substitution a a conversion from base 4 to bas e, confirming with the TI-84

An Indefinite Integral with Exponents (5.5 p 359 #76)

 An Indefinite Integral with Exponents (5.5 p 359 #76)

This includes u-substitution, and an exponential function in base 2

Derivative of a Log in base 2 (5.5 p358 #55)

 Derivative of a Log in base 2 (5.5 p358 #55)

Proving an old Compounded Interest Formula with L'Hôpital's Rule (5.6 p371 #90)

 

Proving an old Compounded Interest Formula with L'Hôpital's Rule (5.6 p371 #90)

We demonstrate the ln technique as well as makeing a product into a ratio so you can use L'Hôpital's Rule.  

Implicit Diff with arctan (5.7 p 380 #71)

 Implicit Diff with arctan (5.7 p 380 #71)

Here we find a tangent line of a function using implicit differentiation, the product rule, the chain rule, and some careful algebra.

Derivatives of Inverse Functions on the TI-84 (5.3 p 340 #71)

 Derivatives of Inverse Functions on the TI-84 (5.3 p 340 #71)

The Derivative of an Inverse Function (5.3 p340 #67)

 The Derivative of an Inverse Function (5.3 p340 #67)

The derivative of a inverse function is the reciprocal of the derivative of the inverse function.  Be mindful of the exchange of values (x,y) to (y,x) with inverse functions

Intersections of Polar Curves (10.5 p 735 #29)

 Intersections of Polar Curves (10.5 p 735 #29)

Limiting the Domain of a Absolute Value Function so it would have an inverse (5.3 p340 #57)

 Limiting the Domain of a Absolute Value Function so it would have an inverse (5.3 p340 #57)

Inverse Functions (5.3 p 340 #53)

 Inverse Functions (5.3 p 340 #53)

We have more than just the horizontal line test now, if a function is strictly monotonic, it will have an inverse.

Celsius/Fahrenheit Inverse Functions (5/3 p 340 #50)

 Celsius/Fahrenheit Inverse Functions (5/3 p 340 #50)

Review of a Midpoint Riemann Sum (5.2 p 331 #77)

 Review of a Midpoint Riemann Sum (5.2 p 331 #77)

The Area of a Region That Involves a Secant Function (5.2 p331 #71)

 The Area of a Region That Involves a Secant Function (5.2 p331 #71)

Another Definite Integral without u-Substitution (5.2 p 331 #65)

 Another Definite Integral without u-Substitution (5.2 p 331 #65)

A Definite Integral Requiring u-Substitution (5.2 p 330 #51)

 A Definite Integral Requiring u-Substitution (5.2 p 330 #51)

Another Slope Field and Differential Equation with Natural Logs (5-2 p330 #50)

 Another  Slope Field and Differential Equation with Natural Logs (5-2 p330 #50)

Slope Fields and Differential Equations with Natural Logs (5-2 p330 #49)

 Slope Fields and Differential Equations with Natural Logs (5-2 p330 #49)

A slope field and a diffyQ checked with the TI-84

A u-Sub Involving Natural Log ln x (5-2 p330 #49)

 A u-Sub Involving Natural Log ln x (5-2 p330 #49)

Long Division AND u-Substitution help this Integral (5-2 p330 #21)

 Long Division AND u-Substitution help this Integral (5-2 p330 #21)

Long Division Before Integrating (5-2 p330 #17)

Long Division Before Integrating (5-2 p330 #17)

This one looks like a difficult u-sub, but turns out to be easy after long division!

Differentiating Using Logarithms Instead of the Quotient Rule (5-1 p322 #79)

 Differentiating Using Logarithms  Instead of the Quotient Rule (5-1 p322 #79)

A Tangent Line Confirmed with the TI-84 (5-1 p322 #69)

 A Tangent Line Confirmed with the TI-84 (5-1 p322 #69)

We confirm our result with the (DRAW)(5:Tangent) feature on the TI-84

Taking the Derivative of a Log Function (5-1 p322 #55)

 Taking the Derivative of a Log Function (5-1 p322 #55)

We use the quotient rule together in this example.

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)