Day 50 - Integration by Substitution - 03.16.16

Update
  • Unit 4 Test
  • Last Day to Remediate Unit 3 Test
    • Next Friday, March 25th!

  1. Evaluate the integral

    1. 24

    2. 180

    3. 1352

    4. –180

    5. 10,170

  2. Evaluate the integral

  3. Find given

  4. Find the average value of on [3, 7].Use the following average value equation to solve:  Submit your answer in "Other".

  5. Determine the area of the indicated region for the function . Submit your answer in "Other". Answer in terms of pi.



Review
  • Differentiation Rules
    • Power Rule
    • Constant Multiple Rule
    • Trigonometric Functions
  • How can antiderivatives be calculated using inverse differentiation rules?
  • How can area under a graph of a function be calculated?

        Lesson

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

          Homework
          • Study!


          Standard(s)
          • 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.