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.
If you have any questions about the course, I can always be contacted at
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- This course has now concluded. Best of luck with your future
studies.
- Assignment 1
(due Friday, September 20th)
- Assignment 2
(due Friday, September 27th)
- Assignment 3
(due Friday, October 4th)
- Assignment 4
(due Friday, October 18th)
- Assignment 5
(due Friday, October 25th)
- Assignment 6
(due Friday, November 1st)
- Assignment 7
(due Friday, November 15th)
- Assignment 8
(due Friday, November 22nd)
- Worksheet on Integrals
Leading to Inverse Trigonometric Functions
- Worksheet on Basic
Integration
- Worksheet on Area
Under a Curve
- Worksheet on Definite
Integrals as Limits of Riemann Sums
- Worksheet on Area
Between Curves
- Worksheet on
Integration Strategies
- Worksheet on Improper
Integrals
- Worksheet on
Differential Equations
- Worksheet on
Exponential Growth and Decay
- Worksheet on
Volumes
- Test 1
(written Wednesday, October 16th)
- Test 2
(written Wednesday, November 13th)
I do not make lecture notes available outside of class; if you miss a
lecture for any reason, I strongly encourage you to borrow the
corresponding notes from a classmate. However, as a guide, here is a brief
summary of what was covered in each lecture.
- September 6th: Antiderivatives, indefinite integrals, elementary
indefinite integrals (Section 1.1)
- September 9th: Basic properties of indefinite integrals, elementary
integrals with linear composition (Section 1.1)
- September 11th: Integrals with non-linear composition (Section
1.1); differentials, the method of substitution (Section
1.2)
- September 13th: Substitution and more complicated integrands,
substitution and non-composite integrands, substitution and rational
integrands, long division of polynomials (Section 1.2)
- September 16th: Integrals of tangent, cotangent, secant and cosecant
functions (Section 1.2); integrals leading to inverse trigonometric
(arcsine, arctangent and arcsecant) functions (Section 1.3)
- September 18th: Completing the square, integrals which lead to inverse
trigonometric functions after completing the square (Section
1.3)
- September 20th: Integration by parts, integrals which require multiple
uses of integration by parts (Section 1.4)
- September 23rd: Criteria for setting up integration by parts, when the
original integral recurs while using integration by parts, combining
integration by substitution and integration by parts (Section
1.4)
- September 25th: Reduction formulas (Section 1.4); the problem
of computing the area under a curve, subintervals and partitions
(Section 2.1)
- September 27th: Lower and upper sums, sigma notation (Section
2.1)
- September 30th: Properties of sigma notation, summation formulas,
computing lower and upper sums with n subintervals (Section
2.1)
- October 2nd: Sample points, area under a curve as the limit of a
Riemann sum, irregular partitions (Section 2.1); the definite
integral (Section 2.2)
- October 4th: Evaluating definite integrals by finding the limit of a
Riemann sum, definite integrals with unusual bounds, basic properties of
definite integrals, the additive property of intervals for definite
integrals (Section 2.2)
- October 7th: The First Fundamental Theorem of Calculus (Section
2.3)
- October 9th: Improper integrals, the First Fundamental Theorem of
Calculus and its interaction with the additive property of definite
integrals, integration by parts and the First Fundamental Theorem,
integration by substitution and the First Fundamental Theoem (Section
2.3)
- October 11th: The Second Fundamental Theorem of Calculus (Section
2.3); computing the area between curves (Section 2.4)
- October 18th: Vertically simple regions, regions comprised of
vertically simple regions, horizontally simple regions (Section
2.4)
- October 21st: Converting a region comprised of vertically simple
regions into a horizontally simple region (Section 2.4)
- October 23rd: Finding area by integrating with respect to y to
simplify the integral, graphing functions of y directly
(Section 2.4); the method of partial fractions, partial fractions
with unique and repeated linear factors in the denominator (Section
3.1)
- October 25th: Partial fractions with unique and repeated irreducible
quadratic factors in the denominator (Section 3.1)
- October 28th: Alternative method for finding the constants in the
partial fraction decomposition, partial fractions and improper rational
functions, partial fractions and definite integrals (Section 3.1);
overall strategy for trigonometric integrals (Section 3.2)
- October 30th: Sine/cosine and tangent/secant integrals (Section
3.2)
- November 1st: Cotangent/cosecant integrals, integrals involving other
trigonometric combinations (Section 3.2); overall strategy for
trigonometric substition (Section 3.3)
- November 4th: Trigonometric substitutions of tangent, sine and
cosecant (Section 3.3)
- November 6th: Trigonometric substitution with u-substitution,
trigonometric substitution and completing the square (Section
3.3); strategies for integration (Section 3.4)
- November 8th: Improper integrals with one or two infinite bounds,
improper integrals with a discontinuity at the lower bound (Section
3.5)
- November 15th: Improper integrals with a discontinuity at the upper
bound or on the interval of integration, improper integrals requiring
l'Hopital's Rule (Section 3.5); differential equations, general
solutions, initial conditions, particular solutions (Section
4.1)
- November 18th: Differential equations and kinematics problems
(Section 4.1); separable equations, exponential growth and decay
(Section 4.2)
- November 20th: Applying the exponential growth/decay equation,
Newton's Law of Cooling (Section 4.2)
- November 22nd: Applying Newton's Law of Cooling (Section 4.2);
solids of revolution, volume by the disc method (Section 4.3)
- November 25th: Volume by the disc method for vertical axes of
revolution, the disc-washer method (Section 4.3)
- November 27th: The disc-washer method for vertical axes of revolution
(Section 4.3); the shell method (Section 4.4)
- November 29th: Choosing between the disc-washer method and the shell
method (Section 4.4)
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