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04 - Integrals



Assessment Standards


4.1      Can compute Riemann sums using left, right, and midpoint evaluation.

4.3 Can apply the Fundamental Theorem of Calculus to compute integrals.
4.5Can use substitution to compute integrals.



 

Chapter 4 - Integrals

 

Areas and Distances

·    Computation of Riemann sums using left, right, and midpoint evaluation.

If we know how something is changing, how can we find out how much the function has changed in total?

Riemann Sums

HW: Using rectangles to approximate area under a curve

HW: Riemann sums and sigma notation


The Definite Integral

·    Definite integral as limit of Riemann sums over equal subdivisions.

·    Basic properties of definite integral, such as additivity, and linearity.

·    Use of Riemann sums to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.

Just as we used the limit to precisely study and define the derivative, we’ll now use it to precisely define the integral – a function that tells us how much something has changed.

Riemann sums and integrals

HW: The definite integral as the limit of a Riemann sum

Properties of Definite Integrals

KHW: Properties of Integrals

The Fundamental Theorem of Calculus

·    The FTC, using the integral of a rate of change to give accumulated change. Use of FTC to evaluate definite integrals.

·    Use of FTC to represent a particular anti-derivative and the analytic and graphical analysis of functions so defined.

How are the derivative and the integral related?


Functions defined by integrals

The Fundamental Theorem of Calculus

KHW: Functions defined as Integrals

KHW: Antiderivatives

KHW: FTC

Indefinite Integrals and the Net Change Theorem

·    Finding specific anti-derivatives using initial conditions, including applications to motion along a line.

·    Interpretation of FTC2 as the total change of a rate function.

·    Representation of an indefinite integral as a family of functions.

Using integrals, we can now find out exactly how much a physical system has changed over time. We’ll see how the initial values of variables describing a system can impact how the system changes over time.

Indefinite Integrals and anti-derivatives

KHW: Indefinite Integrals

The Substitution Rule

·    Anti-derivatives by substitution of variables, including change of limits for definite integrals.

·    The two methods of using substitution to compute definite integrals.

Real world functions are often complicated. We will learn how to simplify functions using substitution, and see how this impacts the system graphically.

The Reverse Chain Rule

Back Substitution

U-Subs with Definite Integrals





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4A - Practice, Practice, Practice! [u4.1: 1, 3, 5, 7, 15, 17, 19] [u4.2: 1, 5, 7, 9, 17, 23, 29, 33, 35, 37, 47, 55, 59] 
4B - yet more practice! [u4.3: 1-47 odd] 
4C [u4.4: 5, 9, 11, 15, 17, 21, 33, 35, 39, 45, 47, 57] 
4D [u4.5: 1-10, 11, 13, 17, 21, 23, 31, 39, 47, 53] 
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Nikhil Joshi,
Apr 23, 2018, 9:24 AM
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Nikhil Joshi,
May 14, 2018, 9:29 AM
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Nikhil Joshi,
May 18, 2018, 9:29 AM
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Nikhil Joshi,
Apr 27, 2018, 9:41 AM
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Nikhil Joshi,
Apr 27, 2018, 9:14 AM