Day 51 - Integration by Substitution - 10.28.15

  • Questions on the Checkpoints from yesterday?

Bell Ringer
  • Find the definite integrals (area of the region) of the following. Be sure to check your answers with your calculator.

  • Differentiation Rules
    • Power Rule
    • Constant Multiple Rule
    • Trigonometric Functions
  • Integration (Antidifferentiation)
    • How can antiderivatives be calculated using inverse differentiation rules?
    • Introduction to Integration (video)


      • Posted on the board at the end of the block.

      • Create questions that you may have on any of the Checkpoints or Extra Practice 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.