Day 51 - Area Under a Curve - 10.29.14

Updates
  • Unit 4 Test will be open note!
  • Unit 4 Test will be on Friday, 11.14.14
Bell Ringer
  • Area Under a Function

  1. Approximate the area under the function f(x) = 2x + 5 on the closed interval [0, 2] using 4 rectangles.

    1. Upper Area = 13; Lower Area = 15

    2. Upper Area = 13; Lower Area = 14

    3. Upper Area = 12; Lower Area = 15

    4. Upper Area = 10; Lower Area = 15

    5. none of the above

  2. Calculate the actual area under the function f(x) = 2x + 5 on the closed interval [0, 2].

    1. 12

    2. 13

    3. 14

    4. 15

    5. none of the above

  3. Approximate the area under the function f(x) = cos(x) on the closed interval [0, π/2] using 4 rectangles.

    1. Upper Area = 1.1835; Lower Area = 0.7908

    2. Upper Area = 0.1835; Lower Area = 1.7908

    3. Upper Area = 2.1835; Lower Area = 0.7908

    4. Upper Area = 1.1835; Lower Area = 1.7908

    5. none of the above

  4. Calculate the actual area under the function f(x) = cos(x) on the closed interval [0, π/2].

    1. 0

    2. 1

    3. 2

    4. π

    5. none of the above
Review
  • Prerequisites
    • Derivatives
      • Derivative Rules
        • Power Rule
        • Constant Rule
      • Trig Derivatives
  • Antiderivative
    • Indefinite Integral
    • Definition
    • Integral Sign
    • Integrand
    • Variable of Integration
    • Constant of Integration
    • General Solution
  • Basic Integration Rules
Lesson
  • Area Under a Function
    Exit Ticket
    • Posted on the board at end of the block.
    Lesson Objective(s)
    • How can the area under a curve be approximated?
    • How can the area under a curve be calculated?
    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.
    Math    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