7 Integration

In this module, you will learn about integration and the Fundamental Theorem of Calculus.

Lessons

• Lesson 7.1: The Area Under a Curve

• Lesson 7.2: Definite Integrals and Properties of Integrals

• Lesson 7.3: Anti-Derivatives and The Fundamental Theorem of Calculus

  • Lesson 7.4: Integrations by Basic (u) Substitutions

Module Start & End Dates

Please check the syllabus for module start & end dates.

Show Your Work Problems

Module 7 assignment

Other Information

Project 7: Integral Functions

Select one problem of the two problems given to do and turn it in.

Problem 1: Let g(x)= ∫^x_-2 (2-|t|) dt where x is in [-2, ∞ ).

• 1. Let the horizontal axis be t and the vertical axis be y. Graph y=f(t)=2 - | t |.

  2. Recall that if the function f(t) ≥ 0 over the interval from t=a to t=b, then the definite integral of f(t) from t=a to t=b gives the area between the graph of f(t) and the t-axis from t=a to t=b. If the function f(t) ≤ 0 on the interval from t=a to t=b, then the definite integral of f(t) from t=a to t=b gives the opposite (negative) of the area between the graph of the f(t) and the t-axis from t=a to t=b. If the function f(t) is positive from t=a to t=c (where c is between a and b) and negative on the interval from t=c to t=b, then, the definite integral of f(t) from t=a to t=b will give you net area= (area between the curve from t=a to t=c) - (area between the curve from t=c to t=b). Now find the following function values: g(- 2), g(- 1), g(0), g( 1), g(2), g(3), g(4).

  3. On what interval(s) is g increasing? Why?

  4. Where does g have a maximum value? Why?

Problem 2: Let the graph of f(t) be given below. Suppose that g(x)= ∫₀^x f(t) dt.

• 1. At what values of x do the local max/min values of g occur? Why?

  2. Where (at what x value) does g attain its absolute maximum value? Why?

  3. On what intervals is the graph of g concave down? Why?