MATH 1241: Calculus II

• Use sigma notation for finite sums and perform algebraic manipulations using this notation.
• Set up and evaluate Riemann sums using left endpoints, right endpoints, or midpoints, and show how these Riemann sums may be interpreted in terms of areas of rectangles.
• Define the definite integral of a function as a limit of a Riemann sum and, conversely, recognize a limit of this form as a definite integral.
• Relate definite integrals to derivatives by the Fundamental Theorem of Calculus.
• Interpret definite integrals as areas and use definite integrals to compute areas.
• Approximate definite integrals by Riemann sums and also by the Trapezoid Rule.
• Evaluate antiderivatives (indefinite integrals) and definite integrals by appropriate algebraic manipulation of the integrand, such as completing the square, polynomial division, partial fraction decomposition, and the use of trigonometric identities.
• Evaluate antiderivatives and definite integrals by the method of substitution, including trigonometric substitution.
• Evaluate antiderivatives and definite integrals by the method of integration by parts.
• Interpret improper integrals as limits of proper integrals and evaluate them accordingly (or show divergence).
• Find areas of regions in the plane bounded by curves, using appropriate integrals.
• Compute the average value of a function on an interval.
• Find the volume of a solid in cases where the volume may be computed by integrating the cross-sectional area of the solid (in particular, for solids that can be generated by rotating a planar region about an axis). Also find volumes using cylindrical shells, when appropriate.
• Set up an integral for the length of a curve defined as the graph of a function.
• Set up and compute an appropriate integral for the work done by a force in various instances where the force is not constant or different parts of the object move different distances. In particular, find the work done against gravity to empty a tank of a specified shape by pumping the contents through an outlet.
• Determine whether a given function is a solution to a given ordinary differential equation or initial-value problem.
• Relate first order differential equations to direction fields.
• Solve first order differential equations exactly by separation of variables when possible.
• Solve problems involving exponential growth and decay.
• Use sigma notation for infinite series and perform algebraic manipulations using this notation.
• Compute sums of convergent geometric series.
• Use the Ratio Test to determine convergence or divergence of numerical series (when this test works) and to determine the radius of convergence for power series.
• Compute Taylor series (or Maclaurin series) using the formula for the coefficients.
• Manipulate power series using various operations, including substitution, differentiation, and integration, and thereby obtain Taylor series without directly calculating the coefficients.

Unit 1: Integrals

Unit 2: Applications of Integration

Unit 3: Differential Equations 

Unit 4: Infinite Series 

There is no required textbook to purchase. All materials will be provided within the course.

A good-quality scientific calculator is required.

To successfully complete this course, students must achieve a passing grade of 50% or higher on the overall course, and 50% or higher on the final mandatory exam.

Note: The final exam for this course is only available as a paper exam and must be taken in person at an approved Testing Centre. Please email exams@tru.ca with any questions.

An Open Learning Faculty Member is available to assist students. Students will receive the necessary contact information at the start of the course.