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Mathematics 1001 is a direct sequel to Mathematics 1000, and serves as an
introduction to integral calculus. It begins by introducing the concept of
the antiderivative and basic techniques for indefinite integration. The
definite integral is then developed by way of Riemann sums and the
Fundamental Theorem of Calculus. Advanced integration techniques are
studied, before the course concludes by considering some applications of
integration (such as simple differential equations, and area and volume
problems). In order to do well, it is essential that students be
comfortable with all aspects of differentiation from Math 1000.
On this page, you'll be able to download course handouts (including
assignments, tests and solutions). If there is a disruption to the class
schedule -- because of weather, for instance -- you should check this page
for news relating to modified due dates and the like. Corrections to any
errors on assignments or worksheets will also be posted here... although
I'll try my best not to make any!
Remember that you can always contact me at
[Your browser cannot view this email address; please turn on JavaScript] .
- This course has now concluded. Best of luck with your future
studies.
- Welcome to Math
1001!
- Week 1: Lecture #1 -- Antiderivatives, indefinite integrals,
strategy for integration, common integrals including the Power Rule for
integrals (Section 1.1)
- Week 1: Lecture #2 -- Common integrals involving transcendental
functions, basic properties of indefinite integrals (Section
1.1)
- Week 1: Lecture #3 -- Elementary integrals with linear
composition, integrals with non-linear composition, the relationship
between derivatives and indefinite integrals (Section 1.1)
- Week 2: Lecture #4 -- Differentials, the method of substitution
(Section 1.2)
- Week 2: Lecture #5 -- Substitution and non-composite integrands,
substitution and rational integrands (Section 1.2)
- Week 2: Lecture #6 -- Long division of polynomials, integrals of
tangent, cotangent, secant and cosecant functions (Section 1.2);
why integrals do not usually lead to arccosine, arccotangent or
arccosecant functiond (Section 1.3)
- Week 3: Lecture #7 -- Integrals leading to inverse trigonometric
(arcsine, arctangent and arcsecant) functions, completing the square
(Section 1.3)
- Week 3: Lecture #8 -- Integrals which lead to inverse
trigonometric functions after completing the square (Section 1.3);
integration by parts (Section 1.4)
- Week 3: Lecture #9 -- Strategy for integration by parts,
integrals which require multiple uses of integration by parts, integration
parts applied in the absence of a product, combining integration by
substitution and integration by parts (Section 1.4)
- Week 4: Lecture #10 -- Guidelines for choosing an integration
technique, reduction formulas (Section 1.4); the problem of
computing the area under a curve (Section 2.1)
- Week 4: Lecture #11 -- Subintervals and partitions, inscribed
and circumscribed rectangles, lower and upper sums (Section
2.1)
- Week 4: Lecture #12 -- Estimating the area under a curve using
n subintervals, sigma notation, summation formulas (Section
2.1)
- Week 5: Lecture #13 -- Applying the summation formulas,
computing lower and upper sums with n subintervals, sample points,
area under a curve as the limit of a sum (Section 2.1)
- Week 5: Lecture #14 -- Irregular partitions, the Riemann sum
(Section 2.1); the definite integral (Section 2.2)
- Week 5: Lecture #15 -- Evaluating definite integrals by finding
the limit of a Riemann sum, definite integrals and area, definite
integrals with unusual bounds, basic properties of definite integrals, the
additive property of intervals for definite integrals (Section
2.2)
- Week 6: Lecture #16 -- The First Fundamental Theorem of
Calculus, applying the First Fundamental Theorem with the Chain Rule,
applying the First Fundamental Theorem when the lower bound of the
definite integral contains a variable, the Second Fundamental Theorem of
Calculus (Section 2.3)
- Week 6: Lecture #17 -- The proof of the Second Fundamental
Theorem of Calculus, applying the Second Fundamental Theorem, improper
integrals, the Second Fundamental Theorem and its interaction with
the additive interval property (Section 2.3)
- Week 6: Lecture #18 -- Integration by parts and the Second
Fundamental Theorem of Calculus, the method of substitution and the Second
Fundamental Theorem (Section 2.3); the area between curves
formula (Section 2.4)
- Week 7: Lecture #19 -- Regions that a naturally bounded by curves,
vertically simple regions, regions comprised of several vertically simple
regions (Section 2.4)
- Week 7: Lecture #20 -- Horizontally simple regions, graphing
functions of y, when to find area by considering horizontally
simple regions (Section 2.4)
- Week 7: Lecture #21 -- Partial fraction decompositions, partial
fractions arising from unique linear factors, repeated linear factors,
unique irreducible quadratic factors, and repeated irreducible quadratic
factors (Section 3.1)
- Week 8: Lecture #22 -- Integration by the method of partial
fractions, partial fractions and definite integrals, partial fractions and
improper rational functions (Section 3.1)
- Week 8: Lecture #23 -- Sine/cosine integrals with at least one odd
power, sine/cosine integrals with only even powers (Section 3.2);
trigonometric substitution (Section 3.3)
- Week 8: Lecture #24 -- Trigonometric substitutions of the tangent,
sine and secant functions (Section 3.3)
- Week 9: Lecture #25 -- Trigonometric substitution and
u-substitution, trigonometric substitution and definite integrals
(Section 3.3); improper integrals with an infinite upper bound
(Sections 3.4)
- Week 9: Lecture #26 -- Improper integrals with a lower infinite bound,
improper integrals with two infinite bounds, infinite regions with finite
area, improper integrals with a discontinuity at the lower or upper
bounds, improper integrals with a discontinuity on the interval of
integration (Section 3.4)
- Week 9: Lecture #27 -- Improper integrals that require
l'Hôpital's Rule (Section 3.4); differential equations,
general and particular solutions, initial conditions, initial value
problems (Section 4.1)
- Week 10: Lecture #28 -- Differential equations and kinematics,
higher-order differential equations (Section 4.1)
- Week 10: Lecture #29 -- Separable differential equations, separable
equations with implicitly-defined solutions, exponential growth and
exponential decay (Section 4.2)
- Week 10: Lecture #30 -- Examples of exponential growth and decay,
applying exponential growth and decay beyond population biology
(Section 4.2); the logistic model of population growth (Section
4.3)
- Week 11: Lecture #31 -- Solutions of the logistic model, dynamics of
the logistic model, carrying capacity, equilibrium points, a model for
constant harvesting (Section 4.3)
- Week 11: Lecture #32 -- Solutions of the constant harvesting model
(Section 4.3); predator-prey models, the Lotka-Volterra equations
and their equilibrium points, models of co-operation and competition
between populations (Section 4.4)
- Week 12: Lecture #33 -- Discrete and continuous random variables,
probability density functions, the mean value of a probability density
function (Section 4.5)
- Week 12: Lecture #34 -- Computing the mean value of a probability
density function (Section 4.5); volumes of solids of revolution,
the Disc-Washer Method (Section 4.6)
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