Chapter 5 - Integrals

Areas and Distances

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

·    Using the integral of a rate of change to find accumulated change.

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

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 the definition of the definite integral

Definition of the Definite Integral

Properties of the Definite integral

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 analytical and graphical analysis of functions so defined.

How are the derivative and the integral related?

The Fundamental Theorem of Calculus

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.

Integral Defined Functions

Net Change Theorem

Displacement vs. Distance

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 analytical substitution, and see how this impacts the system graphically.

Simple Substitutions

Chapter 6 - Applications of Integration

More About Areas

·    Appropriate integrals are used in a variety of applications to model physical, social, or economic situations.

·    Specific application should include finding the area of a region.

The area under a curve and between two curves can be interpreted in many powerful ways. We’ll learn how to apply this simple area concept to very different real world situations.

The Area Between Curves

Volume

·    Finding the volume of a solid with known cross-sections.

·    Volumes of solids of revolution.

We’ll learn how to find the volumes of complex shapes using calculus. These shapes can often times be hard to pictures, so we’ll build real-world models, and 3D computer models to better understand them.

Volumes of Solids of Revolution: Disks and Rings

Volume of a Solid with Known Cross-Section

Volumes by Cylindrical Shells (Optional)

·    The method of cylindrical shells to compute volumes of revolution.

·    Comparisons between the shell and washer methods.

We’ll use the natural symmetries of rotation to find the volumes of even more solids.

Volumes of Solids of Revolution: Cylindrical Shells

·    Computing the amount of work (in the Physics sense) done in a system.

The Average Value of a Function

·    Finding the average value of a function and the Mean Value Theorem.

·    A geometric interpretation of the MVT for integrals.

We know how to find the average rate of change of a function. What does it mean to find the average value of a function?

The Mean Value Theorem for Integrals

Center of Mass and Density