Engineering and Technology Calculus II

Course Description

EGN3046 – Engineering and Technology Calculus II is a 3-credit-hour applied calculus course designed specifically for engineering technology programs and applied engineering contexts. As the continuation of EGN2045 (Engineering and Technology Calculus I), the course extends applied calculus methods to integration techniques, applications of integration, sequences and series, and an introduction to differential equations — providing the calculus tools needed for upper-division engineering technology coursework. Distinct from the standard engineering calculus sequence (MAC2311, MAC2312, MAC2313) taken by engineering majors, EGN3046 emphasizes applied methods with reduced theoretical depth.

Topics typically include integration techniques (substitution, integration by parts, partial fractions, trigonometric substitutions), applications of integration (area, volume, work, fluid pressure, center of mass), improper integrals, sequences and series (with applications to engineering — Taylor series, Fourier series introduction), and an introduction to ordinary differential equations. The applied emphasis means greater focus on calculation, problem-solving, and engineering applications rather than the theorem-and-proof structure of MAC2312.

EGN3046 is a Florida common course offered at approximately 2 Florida institutions. Together with EGN2045, it provides the applied calculus foundation for engineering technology programs (B.S. or B.A.S. in Engineering Technology). Students intending to transfer to standard engineering programs should consult target institutions about whether EGN3046 satisfies the calculus II requirement (it typically does not articulate as MAC2312 for engineering programs). EGN3046 transfers as the equivalent course at all Florida public postsecondary institutions per SCNS articulation policy where the receiving institution accepts the course.

Learning Outcomes

Required Outcomes

Upon successful completion of this course, students will be able to:

• Apply integration techniques, including substitution (u-substitution); integration by parts (∫u dv = uv - ∫v du); partial fractions decomposition for rational integrands; trigonometric substitutions; the choice of method.
• Apply integration applications — area, including the area between curves; the proper choice of integration variable; engineering applications.
• Apply integration applications — volume, including volume by cross-sections; volume of solids of revolution (disk method, washer method, shell method); the choice of method based on geometry.
• Apply integration applications — engineering quantities, including work (∫F·dx for a variable force); fluid pressure on submerged surfaces; centroids and centers of mass; engineering applications.
• Apply improper integrals, including integrals with infinite limits; integrals with infinite integrand; convergence and divergence; engineering applications.
• Apply sequences, including the limit of a sequence; common types of sequences; convergence of sequences; engineering applications.
• Apply infinite series, including geometric series; the divergence test; the integral test; the comparison tests; the ratio test; the root test; absolute and conditional convergence.
• Apply power series, including the radius and interval of convergence; representation of functions as power series; engineering applications.
• Apply Taylor series and Maclaurin series, including the Taylor expansion of common functions (e^x, sin x, cos x, ln(1+x)); the use of Taylor series for approximation; engineering applications.
• Apply introductory ordinary differential equations, including separable equations; linear first-order equations; engineering applications (mixing problems, RC circuits, growth and decay).
• Apply introductory second-order linear ODEs with constant coefficients at conceptual level (where included), with engineering applications (mechanical vibrations, RLC circuits) at introductory level.
• Apply computational tools (graphing calculator, MATLAB, Python, or Excel) for calculus problem-solving and visualization where appropriate.

Optional Outcomes

• Apply introductory Fourier series (where included), with engineering applications in signal analysis and the analysis of periodic forcing.
• Apply numerical integration at introductory level, including the trapezoidal rule and Simpson's rule.
• Apply introductory partial differentiation at conceptual level (where included).
• Apply calculus to specific engineering technology contexts (mechanical engineering technology, electronic engineering technology, civil engineering technology, etc.) at applied level.

Major Topics

Required Topics

• Review of Differentiation and Basic Integration: Brief review of derivatives and basic integration from EGN2045; the foundation for the more advanced techniques covered in this course.
• Integration by Substitution: The u-substitution method; choosing u; the proper handling of differentials; engineering applications.
• Integration by Parts: The formula ∫u dv = uv - ∫v du; choosing u and dv; the LIATE mnemonic for choosing u; integration by parts twice; engineering applications.
• Partial Fractions: The decomposition of rational functions; distinct linear factors, repeated linear factors, distinct quadratic factors; the method of undetermined coefficients; the application to integration.
• Trigonometric Integrals: Powers of sine and cosine; products of sine and cosine; secant and tangent integrals; the choice of method.
• Trigonometric Substitution: Substitutions for √(a²-x²), √(a²+x²), √(x²-a²); the role of the appropriate substitution; engineering applications.
• Improper Integrals: Integrals with infinite limits (∫_a^∞ f(x) dx as the limit of definite integrals); integrals with infinite integrand; convergence and divergence; engineering applications (Laplace transforms at conceptual level).
• Area Between Curves: The area as ∫[upper - lower] dx or ∫[right - left] dy; the choice of integration variable; the engineering applications.
• Volume by Cross-Sections: Volume as ∫A(x) dx where A(x) is the cross-sectional area; engineering applications (volumes of irregular shapes).
• Volume of Solids of Revolution: The disk method (V = π∫[r(x)]² dx); the washer method (V = π∫[R(x)² - r(x)²] dx); the shell method (V = 2π∫x f(x) dx); the choice of method based on the axis of rotation and the nature of the region; engineering applications.
• Work as an Integral: Work done by a variable force (W = ∫F(x) dx); engineering applications (springs, pumping fluids, lifting objects).
• Fluid Pressure: The hydrostatic pressure (P = ρgh); fluid force on submerged surfaces; engineering applications.
• Centroids and Centers of Mass: The centroid of a planar region; the centroid by integration; the relationship to centroids of composite shapes from statics; engineering applications.
• Sequences: The limit of a sequence; common types of sequences (arithmetic, geometric, Fibonacci); convergence and divergence.
• Series — Foundations: The infinite series concept; partial sums; convergence and divergence; geometric series (∑ar^n converges to a/(1-r) for |r| < 1); the divergence test.
• Series Convergence Tests: The integral test; the comparison test; the limit comparison test; the ratio test; the root test; the alternating series test; absolute and conditional convergence.
• Power Series: The radius of convergence; the interval of convergence; representation of functions as power series; differentiation and integration of power series.
• Taylor and Maclaurin Series: The Taylor formula (f(x) = ∑f^(n)(a)/n! · (x-a)^n); the Maclaurin series (Taylor series at a = 0); common Maclaurin series (e^x, sin x, cos x, 1/(1-x), ln(1+x)); the use of Taylor series for function approximation; engineering applications.
• Introductory Differential Equations: What an ODE is; first-order separable equations; linear first-order equations (the integrating factor μ(x) = e^(∫P(x)dx)); engineering applications (mixing problems, RC circuits, exponential growth and decay, Newton's law of cooling).
• Introductory Second-Order ODEs (Where Included): Constant-coefficient linear ODEs (ay'' + by' + cy = 0); the characteristic equation; the three cases (real distinct, repeated, complex roots); applications to mechanical vibrations and RLC circuits at introductory level.

Optional Topics

• Fourier Series: The Fourier series for periodic functions; Fourier coefficients; convergence; engineering applications in signal analysis.
• Numerical Integration: The trapezoidal rule; Simpson's 1/3 rule; the application to engineering data analysis.
• Introduction to Partial Derivatives: Functions of several variables; partial derivatives at conceptual level; engineering applications (where included).
• Discipline-Specific Applications: Mechanical engineering technology (motion problems, work, fluid statics applications); electronic engineering technology (signal analysis, circuit response); civil engineering technology (structural calculations).

Resources & Tools

• Common Texts: Calculus: An Applied Approach (Larson — continuation from EGN2045); Brief Calculus: An Applied Approach (Larson); Applied Calculus (Hughes-Hallett); Technical Calculus with Analytic Geometry (Kuhfittig — designed specifically for engineering technology); applied calculus textbooks designed for engineering technology programs
• Online Platforms: WebAssign (Cengage — paired with Larson); MyMathLab (Pearson)
• Software: Graphing calculators (TI-84, TI-Nspire); Desmos (free graphing); MATLAB or Python for engineering applications; Excel for tabular calculation
• Reference Resources: Khan Academy Calculus (free); Paul's Online Math Notes (tutorial.math.lamar.edu, free); MIT OpenCourseWare 18.01 Single Variable Calculus (free)

Career Pathways

EGN3046 supports career pathways in engineering technology and applied engineering work — see EGN2045 for the comprehensive list of career pathways. EGN3046 specifically supports advanced engineering technology coursework that requires the integration techniques and series content covered here (signal analysis, advanced circuit analysis, structural analysis, manufacturing process analysis).

Special Information

Engineering vs. Engineering Technology

EGN3046, like EGN2045, reflects the distinction between engineering and engineering technology programs. Students intending to pursue an engineering bachelor's degree at a Florida State University System institution should consult target programs about the calculus requirement; most engineering programs require the standard MAC2311/MAC2312/MAC2313 sequence, and EGN3046 typically does not satisfy this requirement.

The EGN2045/EGN3046 Sequence

Together with EGN2045 (Engineering and Technology Calculus I), EGN3046 provides the applied calculus foundation for engineering technology programs. Students completing this sequence have the calculus tools needed for upper-division engineering technology coursework.

General Education and Transfer

EGN3046 is a Florida common course number that transfers as the equivalent course at all Florida public postsecondary institutions per SCNS articulation policy where the receiving institution accepts the course. Students should verify articulation with the receiving institution before relying on transfer credit.

Course Format

EGN3046 is offered in face-to-face, hybrid, and online formats. Mathematics courses translate well to online delivery; many institutions offer fully online sections.

Position in the Engineering Technology Curriculum

EGN3046 is typically taken in the second or third year of engineering technology study, after EGN2045. The course is followed by upper-division engineering technology coursework that applies the calculus tools developed here.

Prerequisites

EGN3046 typically requires:

• EGN2045 (Engineering and Technology Calculus I) with grade of C or better
• College-level reading placement

Students should have current proficiency in differentiation and basic integration before beginning EGN3046.