Day 56 - Integration by Substitution - 11.05.14

  • Unit 4 Test will be open note!
  • Unit 4 Test will be on Friday, 11.14.14
  • Hand back App of Diff Test
    • Average = 83.6%

Bell Ringer
  • Fundamental Theorem of Calculus
  1. Find the area under the function f(x) = 2x on the closed interval [3, 6] using the limit definition.

    1. -27

    2. 36

    3. 9

    4. 27

    5. none of the above

  2. Evaluate

    1. (1/3)(x2 + 1)3

    2. (1/3)(x2 + 1)3 + C

    3. (x2 + 1)3 + C

    4. -(1/3)(x2 + 1)3

    5. none of the above

  • Prerequisites
    • Derivatives
      • Derivative Rules
        • Power Rule
        • Constant Rule
      • Trig Derivatives
    • Sigma Notation
  • Antiderivative
    • Indefinite Integral
    • Definition
    • Integral Sign
    • Integrand
    • Variable of Integration
    • Constant of Integration
    • General Solution
  • Basic Integration Rules
  • Area Under a Function
  • Fundamental Theorem of Calculus


      Exit Ticket
      • Posted on the board at end of the block.
      Lesson Objective(s)
      • How can the Fundamental Theorem of Calculus be used to solve problems?

      • APC.10
        • Use Riemann sums and the Trapezoidal Rule to approximate definite integrals of functions represented algebraically, graphically, and by a table of values and will interpret the definite integral as the accumulated rate of change of a quantity over an interval interpreted as the change of the quantity over the interval
        • Riemann sums will use left, right, and midpoint evaluation points over equal subdivisions.
      • APC.11
        • ​The student will find antiderivatives directly from derivatives of basic functions and by substitution of variables (including change of limits for definite integrals).
      • APC.12
        • ​The student will identify the properties of the definite integral. This will include additivity and linearity, the definite integral as an area, and the definite integral as a limit of a sum as well as the fundamental theorem.
      • APC.13
        • ​The student will use the Fundamental Theorem of Calculus to evaluate definite integrals, represent a particular antiderivative, and facilitate the analytical and graphical analysis of functions so defined.
      • APC.14
        • ​The student will find specific antiderivatives, using initial conditions (including applications to motion along a line). Separable differential equations will be solved and used in modeling (in particular, the equation y' = ky and exponential growth).
      • APC.15
        • ​The student will use integration techniques and appropriate integrals to model physical, biological, and economic situations. The emphasis will be on using the integral of a rate of change to give accumulated change or on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. Specific applications will include
          • a)​ the area of a region;
          • b)​ the volume of a solid with known cross-section;
          • c)​ the average value of a function; and
          • d) ​the distance traveled by a particle along a line.

      ematical Practice(s)
      • #1 - Make sense of problems and persevere in solving them
      • #2 - Reason abstractly and quantitatively
      • #5 - Use appropriate tools strategically
      • #6 - Attend to precision
      • #8 - Look for and express regularity in repeated reasoning

      Past Checkpoints