The attainment of the objective requires an understanding of the basic concepts of integral calculus: the primitive, the indefinite integral and the definite integral, the improper integral, power series. The focus will be exclusively on real functions of one real variable: algebraic, exponential, logarithmic, trigonometric and inverse trigonometric functions.
The operational aspect of integral calculus will be given priority over the formal. Emphasis will be placed on rigour in reasoning and in the application of methods. It is suggested that the major steps in the development of integral calculus be placed in their historical context. Some of the problems will be taken from other science courses in the program.
Students will be required to use a scientific calculator and must have access to relevant software packages, such as those employed to plot curves and use symbolic computation.
Limit: indeterminate forms, Hospital's rule.
Standard rules and techniques of integration.
Properties of the indefinite integral and the definite integral.
Calculation of lengths, areas and volumes.
The fundamental theorem of calculus.
Separable differential equations.
Taylor and MacLaurin series.
Appropriate use of concepts.
Adequate two- and three-dimensional representations of surfaces and solids of revolution.
Use of algebraic operations in conformity with rules.
Correct choice and application of rules and techniques of integration.
Accuracy of calculations.
Justification of steps in the solution.
Correct interpretation of results.
Use of appropriate terminology.
COMPETENCY: To apply the methods of integral calculus to the study of functions and problem-solving.
Specific Elements of the Competency
Standard of Performance: The student must be able to:
1. To determine the indefinite integral of a function.
1.1 INVERSE TRIGONOMETRIC FUNCTIONS
1.1.1 give the definitions, domains and ranges of the arcsin and arctan functions.
1.1.1 calculate their derivatives (the other four functions are optional).1.2 ANTIDERIVATIVES
1.2.1 define antiderivative, indefinite integral, constant of integration.
1.2.2 give basic indefinite integrals involving algebraic, trig, log, exponential and inverse trig functions (i.e. those following directly from the derivatives).
1.2.3 find antiderivatives satisfying certain boundary conditions.
1.3 DIFFERENTIALS
1.3.1 give the definition of the differential.
1.3.2 use the calculus of differentials.
1.3.3 give the geometric interpretation of the differential and use it to make linear approximations.
1.4 TECHNIQUES OF INTEGRATION
1.4.1 perform algebraic substitutions 1.4.2 integrate trig functions, their powers and combinations, using trig identities where necessary.
1.4.3 find integrals involving or yielding exponential and log functions.
1.4.4 find integrals using integration by parts, partial fractions, completing the square, trig substitution or a combination of these.
1.4.5 find integrals involving or yielding inverse trig functions.
1.4.6 use a table of integrals (optional).
2. To calculate the definite integral and the improper integral of a function in an interval.
2.1 THE DEFINITE INTEGRAL
2.1.1 calculate a Riemann sum using summation formulas.
2.1.2 give the definition of the definite integral and evaluate a definite integral using a Riemann sum.
2.1.3 demonstrate an understanding of the definite integral as an accumulation of infinitesimal quantities.
2.1.4 give the properties of the definite integral.
2.1.5 interpret the definite integral for positive and negative functions.
2.1.6 demonstrate an understanding of both versions of the Fundamental Theorem of Calculus.
2.1.7 calculate a definite integral using the Fundamental Theorem of Calculus, including an integral involving a change of variable (and new limits of integration).
2.1.8 demonstrate an understanding of the Mean Value Theorem for definite integrals.
2.1.9 use the techniques of integration listed above to evaluate definite integrals.
2.2 IMPROPER INTEGRALS
2.2.1 determine the divergence or convergence of an integral where at least one limit is not a real number.
2.2.2 determine the divergence or convergence of an integral which has a discontinuity within the limits of integration.
2.2.3 do likewise for an integral that is a combination of both types above.
3. To calculate the limits of indeterminate forms.
3. FINDING LIMITS OF INDETERMINATE FORMS: L'HOPITAL'S RULE
3.1 demonstrate an understanding of L'Hopital's Rule and the conditions under which it applies.
3.2 use it to find limits of the form 0/0 or ¥/¥.
3.3 recognize the indeterminate forms 0×¥, ¥-¥, 1¥, 00, and ¥0 and evaluate limits of these types by an appropriate method.
4. To calculate volumes, areas and lengths and draw two- and three-dimensional representations. To express concrete problems as differential equations and solve simple differential equations.
4. APPLICATIONS OF THE DEFINITE INTEGRAL
4.1 find the area of a region in the plane using horizontal or vertical slices and judge which method is most efficient for a particular problem.
4.2 find the mean value of a function on an interval.
4.3 find volumes of revolution using the method of disks or cylindrical shells and judge which method is most efficient for a particular problem.
4.4 calculate arclength (optional).
4.5 calculate surface area (optional).
4.6 calculate work and pressure (optional).
4.7 mathematically formulate a situation involving a differential equation.
4.8 solve separable differential equations including exponential growth and decay problems.
5. To analyze the convergence of series.
5. SEQUENCES AND SERIES
5.1 recognize arithmetic and geometric sequences and find a formula for the nth term.
5.2 find a formula for the nth term of a given sequence.
5.3 determine the divergence or convergence of a given sequence.
5.4 use S notation to write a sum in closed form.
5.5 expand a sum written in S notation.
5.6 state and apply the properties of the S notation
5.7 state and apply the formulas for Sn, Sn2, Sn3.
5.8 define the convergence or divergence of an infinite series.
5.9 determine the convergence or divergence of a geometric series and find the sum if convergent.
5.10 apply various tests for convergence (such as the divergence test, the integral test, the comparison tests and the ratio test)(optional).
5.11 find Taylor and MacLaurin series for sin x, cos x, ex, arctan x, ln(1 + x) and related functions.
5.12 estimate remainder, determine radius of convergence (optional)
5.13 perform numerical integraton using the Trapezoid Rule, Simpson's Rule or an appropriate Taylor series (optional).
5.14 use Newton's Method for solving equations, use Picard's Method (optional).
At the teacher's discretion, appropriate and closely related definitions, derivations, proofs and applications using pertinent technology may be added and form part of the evaluation.