Name:________________________
AP Calculus
Unit 3
Notebook Guide
Unit Topic: Integrals (Ch 6) and Finding Antiderivatives (Ch 7)
Date
Lesson Objectives
Textbook Section Homework Assignment
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1. The Definite Integral Again
6.1 HW #1 (p. 304) 1, 3, 5, 7, 9, 11, 13, 15, 17
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2. Fns Defined by Integrals; Accumulation Fns 6.2 HW #2 (p. 312) 1, 3, 6, 7, 9, 11
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3. The Fundamental Theorem Again
6.3 HW #3 (p. 318) 1, 3, 5, 7, 10, 13
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4. Areas of Plane Regions
6.4 HW #4 (p. 324) 1, 5, 7, 9, 15, 17, 21
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5. Antiderivatives
7.1 HW #5 (p. 332) 1, 5, 7, 11, 12, 15, 16, 21, 24, 29, 31, 35
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6. Integration Using the Chain Rule
7.2 HW #6 (p. 336) 1, 3, 5, 7, 9, 13, 17, 21, 25, 29, 31, 33, 35, 37
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7. The Method of Substitution
7.3 HW #7 (p. 342) 3, 5, 9, 13, 19, 21, 25, 27, 33
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8. The Trigonometric Fns and Their Inverses
7.5 HW #8 (p. 354) 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27
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9. Numerical Integration
7.6 HW #9 (p. 363) 1, 3, 5, 7, 9, 10, 13, 17, 18, 21
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10. Review
Unit 3 Review
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11. Problem Solving
Unit 3 Problem Solving
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12. Unit 3 Test
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Note: Please label your homework assignment as shown above along with your name.
Grade Tracking
Homework Quiz #1
/6
Homework Quiz #2
/6
Homework Quiz #3
/6
Homework Quiz #4
/6
Multiple Choice Practice
/6
Problem Solving
/12
Unit Test
/100
Points Earned
/142
Unit Average (w/o bumps) ______
Homework Bumps
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Unit Average (w/ bumps)
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Unit 3 Concepts - Integrals (Ch 6) and Finding Antiderivatives (Ch 7)
An accumulation function is a function that is created from a continuous function using the definite integral. The
function A(x) is called an accumulation function because it is evaluated by accumulating the “area” under the graph of
the integrand function f (t) from a fixed lower limit a to a variable upper limit x. If f (t) is nonnegative over the interval
a ≤ t ≤ x, then A(x) is the area under the graph of f (t) from a to x. If f (t) takes on both positive and negative values over
the interval, then A(x) is the total area of the regions above the x-axis minus the total area of the regions below the
x-axis.
The Second Fundamental Theorem of Calculus states that if f (t) is a continuous function on some interval a ≤ t ≤ x,
and A(x) is the accumulation function equal to the integral of f (t) with respect to t over the interval [a, x], then the
derivative of A(x) is equal to f (x). As with the (first) Fundamental Theorem of Calculus, we see a connection between
derivatives and integrals. In this case, the theorem says that if we integrate a continuous function f, and then
differentiate the result, we return to the original function f.
The method of substitution is used in calculus to transform one integral (antiderivative) into another, the second being
simpler that the first. In general, substitution is a problem-solving strategy used throughout mathematics in which a
given expression is transformed into an equivalent expression that is easier to work with. The steps in using
substitution to find antiderivatives are to (1) choose a substitution function u, determine du, and substitute both into the
integral to replace the original variables, (2) antidifferentiate in terms of u, and (3) resubstitute for u to obtain an
antiderivative in terms of the original variable. When using substitution to evaluate definite integrals, step 1 will also
require the lower and upper limits of integration to be restated in terms of u, and step 3 is unnecessary.
The Midpoint Rule and the Trapezoid Rule are methods of approximating the definite integral of a continuous
function. Recall that the definite integral is defined to be the limiting value of Riemann sums, estimates of the area
under a curve approximated by subdividing the interval and summing the areas of each rectangular region produced.
The Midpoint Rule uses rectangular regions with the height determined by the function value at the midpoint of each
subinterval. The Trapezoid Rule uses the average of the function values at the left and right ends of each subinterval,
with each area now in the shape of a trapezoid.