Problem-solving recipes
A step-by-step procedure for every problem type this course teaches — what to do first, what to do next, and how to tell which case you're in.
What are you being asked to do?
Start with what kind of problem is actually in front of you.
An integral that's a product of two dissimilar functions -- integrate by parts. If the same split has to repeat several times on a polynomial factor, tabular integration organizes it into one pass.
An integral with \sqrt{a^2-x^2}, \sqrt{a^2+x^2}, or \sqrt{x^2-a^2} under the radical -- trigonometric substitution.
An integral of a rational function -- partial fractions. Distinct linear factors alone -> the distinct-factors recipe. A repeated factor or an irreducible quadratic anywhere in the denominator -> the extended recipe that adds those two cases.
An improper integral (an infinite limit, or a vertical asymptote inside the bounds) -- the infinite-limit recipe for an infinite bound, extended by the discontinuity recipe once the bad point is a finite x-value instead.
The area of a region between two curves -- set up the integral first (this decides dx vs. dy and finds the bounds), a separate step from actually integrating.
The volume of a solid of revolution -- the volume-method recipe decides disk, washer, or shell and sets up the integral in one pass.
The length of a curve y=f(x) -- the arc length recipe.
A differential equation to solve -- check whether it separates first. A separable equation with no initial condition -> the separable-equation recipe. With an initial condition -> the extended initial-value-problem recipe. An equation that's linear but doesn't separate -> the integrating-factor recipe instead.
The area of a polar region -- one curve's own interior -> the single-curve recipe. Area shared by, or between, two polar curves -> the extended recipe, which also handles pole intersections.
The radius or interval of convergence of a power series -- the radius recipe finds R and the open interval; the extended recipe adds the two endpoint tests to finish the closed-form interval.
A Taylor polynomial or full Taylor/Maclaurin series -- the finite-degree polynomial recipe, extended into the infinite series once the derivative pattern is clean enough to write a general term.
Whether an improper integral with no elementary antiderivative converges -- the comparison-test recipe (Chapter 3).
Integrating a function with no elementary antiderivative at all -- the series-integration recipe (Chapter 10), once its power series is already known.
Applications of integration
Techniques of integration
Integrate by parts with tabular integration
Use when: the same by-parts split has to be applied several times in a row, because one factor (usually a polynomial) needs repeated differentiation before it reaches 0
Goal: the finished antiderivative in one pass, without hand-tracking u, v, du, dv through several separate applications
Repeated parts and tabular integration
Extends integrate-by-parts once the same split has to repeat several times.
Integrate a rational function with repeated or irreducible quadratic factors
Use when: the denominator has a repeated linear factor (x-r)^k, an irreducible quadratic factor, or both, once distinct linear factors alone (recipe above) no longer cover every factor
Goal: every factor decomposed correctly, using extra terms for the repeated powers and a linear numerator for each irreducible quadratic, then integrated
Partial fractions — repeated and irreducible factors
Extends the distinct-factors recipe once a repeated or irreducible quadratic factor is also present.
Evaluate an improper integral with an infinite discontinuity
Use when: the integrand has a vertical asymptote at an endpoint or interior point of the interval of integration
Goal: a convergence verdict for each piece touching the discontinuity, and the integral's exact value when every piece converges
Improper integrals with infinite discontinuities
Extends the infinite-limit recipe once the bad point is a finite x-value instead.
Differential equations
Solve an initial value problem
Use when: a differential equation is paired with a specific value y(x_0)=y_0, and a single particular solution, not the whole family, is wanted
Goal: the one member of the general solution's family that passes through the given point
Initial value problems and particular solutions
Extends the separable-equation recipe once a starting value pins down the constant.
Polar coordinates
Find the area enclosed by two overlapping polar curves
Use when: the region shared by, or between, two polar curves is needed, not just one curve's own interior
Goal: the total shared or between-curve area, built from separate sector-area integrals over each angle range where a different curve is the boundary
Area Between Polar Curves and Finding Intersections
Extends the single-curve recipe to two overlapping curves, including pole intersections.
Power series and Taylor series
Test the endpoints and complete the interval of convergence
Use when: the radius R and open interval (c-R,c+R) have already been found (recipe above), and the two endpoints still need to be resolved
Goal: the complete interval of convergence, with each endpoint correctly included or excluded
Testing the endpoints
Extends the radius recipe with the two endpoint tests, completing the interval.
Build a full Taylor or Maclaurin series
Use when: the pattern in a function's derivatives at the center is clean enough to write a general n-th term, not just a fixed finite-degree polynomial
Goal: the infinite series sum f^(n)(a)/n! (x-a)^n, in closed sigma-notation form
Taylor and Maclaurin series
Extends the finite polynomial recipe into the full infinite series.