Rutgers Math 152 (01:640:152) Calculus II: Study Guide and Practice
A study guide for Rutgers Math 152, Calculus II for the Mathematical and Physical Sciences: integration, series, power series, polar and complex numbers.
Rutgers University Updated 3 min read
At a glance
- Course code
- 01:640:152 (Math 152)
- Credits
- 4
- Offered
- Fall, spring and summer
- Prerequisites
- Calculus I: Math 135 or Math 151 (not Math 130)
- Covers
- Integral calculus and its applications, infinite series and power series, parametric curves, polar coordinates, complex numbers
- Next course
- 01:640:251 Multivariable Calculus
- Practice here
- Calculus 2
On this page
Rutgers Math 152 (01:640:152, Calculus II for the Mathematical and Physical Sciences) is the second semester of the Rutgers New Brunswick calculus sequence for math, science and engineering majors. It follows Math 151 (or Math 135) and leads to Math 251 Multivariable Calculus. This guide maps every Math 152 topic to a lesson you can practice with instant feedback.
What Rutgers Math 152 covers
The Rutgers math department lists integral calculus and its applications, the theory of infinite series and power series, parametric curves, polar coordinates, and complex numbers. Its learning goals include using integrals to find volumes, arc lengths and surfaces of revolution; finding antiderivatives with u-substitution, integration by parts and trigonometric substitution; and deciding whether a series converges, then finding and using Taylor series.
Like Math 151, it meets for two lectures and one workshop each week (other than asynchronous online sections).
That is a full Calculus 2, with complex numbers as an extra. It lines up with all four units of the Calculus 2 path on PracticeCalculus.
Math 152 topic map
| Math 152 topic | Practice | Read |
|---|---|---|
| u-substitution | U-substitution refresher | U-substitution, the rule |
| Integration by parts | Repeated parts and LIATE | Integration by parts |
| Trig substitution | Trig substitution | Trig substitution |
| Volumes | Volumes with washers | Volumes of revolution |
| Arc length and surfaces of revolution | Arc length and surface area | Improper integrals |
| Infinite series | Series and partial sums | Geometric series |
| Convergence tests | Comparison tests | Convergence tests, ratio test |
| Power and Taylor series | Taylor polynomials | Power series, Taylor series |
| Parametric curves | Arc length of parametric curves | Implicit differentiation |
| Polar coordinates | Polar coordinates | Area between curves |
| Complex numbers | Common Maclaurin series | Taylor series |
We do not have a dedicated complex numbers lesson. The Maclaurin series lesson is the closest match, since Euler's formula e^{i\theta} = \cos\theta + i\sin\theta comes from comparing the series for e^x, \sin x and \cos x.
The Rutgers math and physical sciences sequence
| Course | Credits | Covers |
|---|---|---|
| Math 151 Calculus I | 4 | Derivatives, Mean Value Theorem, intro to integrals |
| Math 152 Calculus II | 4 | Integration, series, parametric, polar, complex numbers |
| Math 251 Multivariable Calculus | 4 | Three-dimensional geometry, partials, multiple integrals, vector analysis |
Students who started with Math 135 may also take Math 152, although the department recommends the Math 151 route.
Example problems at the Math 152 level
1. Integration by parts with a log. With u = \ln x and dv = x^2\,dx:
2. A parametric slope. For x = t^2, y = t^3:
3. Complex numbers in polar form. Since 1 + i = \sqrt{2}\,e^{i\pi/4},
How to do well in Math 152
- Make LIATE a habit. Choosing u well is most of integration by parts. The repeated parts and LIATE lesson drills it.
- Work series problems in mixed sets. Exams rarely tell you which test to use. Practice choosing a test.
- Connect polar and complex numbers. Both use r and \theta, and multiplying complex numbers in polar form is just multiplying lengths and adding angles.
- Keep the formula sheet close, review common calculus mistakes, and take timed practice exams.
Tip: if the terms of a series do not go to zero, it diverges. Check the nth-term test before anything else.
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Integration by parts
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