Day 57 - Fundamental Theorem of Calculus - 04.09.15

  • N/A


Bell Ringer
  • Find the area of the region bounded by the graph of f(x) = x3, the x-axis and the interval [0,4] using 4 rectangles.

  • Prerequisites
    • Derivatives
      • Derivative Rules
      • Power Rule
      • Constant Rule
      • Trig Derivatives
  • Antiderivative (video)
    • Indefinite Integral
    • Definition
    • Integral Sign
    • Integrand
    • Variable of Integration
    • Constant of Integration
    • General Solution
    • Basic Integration Rules
  • Summation (Sigma Notation) (video)
  • Area Under a Curve (video 1) (video 2) (video 3)

  • Fundamental Theorem of Calculus
  • Definite Integrals

      Exit Ticket
      • Posted on the board at the end of the block
      Lesson Objective(s)
      • How can the area under a curve be calculated?
          1. Calculating the area under a curve using summation.

                                                          In-Class Help Requests

                                                          • 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.

                                                          Past Checkpoints