Core idea: the definite integral of a rate of change gives the net change in the original quantity.
- v(t) = s′(t) (velocity is derivative of position)
- a(t) = v′(t) (acceleration is derivative of velocity)
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$s(b)-s(a)=\displaystyle\int_a^b v(t),dt$ → this is displacement (net change in position), signed. -
Total distance traveled (not the same as displacement when velocity changes sign):
$$\text{distance}=\int_a^b |v(t)|,dt$$ - Find where v(t) = 0 to break the interval into pieces where v doesn't change sign, integrate |v| on each piece (flip sign of the integral where v<0), and add the absolute values.
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$v(b)-v(a)=\displaystyle\int_a^b a(t),dt$ → net change in velocity.
If R(t) is a rate (of water flow, population growth, cost, income, etc.):
- If R(t) can be inflow minus outflow, or two competing rates, integrate the difference of the rates to get net change.
- To find the amount at time b, given amount at time a:
$$\text{amount}(b) = \text{amount}(a) + \int_a^b R(t),dt$$ - Read carefully whether the problem wants net change (signed, can cancel out) or total accumulation (use |R(t)|, similar to total distance).
Common exam setup: given v(t) on [a,b], compute (i) displacement, (ii) total distance traveled — know both formulas and when each applies.
Always graph the region first (and the axis of revolution) — this alone is worth points.
Used when the region touches the axis of revolution directly (no gap).
Used when there's a gap between the region and the axis (revolving the area between two curves).
- R(x) = outer radius (farther curve from axis), r(x) = inner radius (closer curve to axis). Outer minus inner, squared first, same order as polar area.
- If the axis of revolution is not the x- or y-axis (e.g., y = k or x = h), adjust the radius: R(x) = f(x) − k, etc. — always distance from curve to axis, not the curve's value itself.
Used when it's easier to integrate along the axis parallel to the axis of revolution (avoids solving for x in terms of y, or vice versa).
- the factor (x or y) = radius = distance from the representative strip to the axis of revolution (adjust with ± shift if axis isn't x=0 or y=0),
- h(x) (or h(y)) = height of the shell = length of the strip = (top curve − bottom curve), or (right curve − left curve).
- Identify the axis of revolution and whether the region is more naturally described as a function of x or of y.
- If slicing perpendicular to the axis gives simple radius/radii → disk or washer.
- If slicing perpendicular to the axis requires solving for the "wrong" variable, or the region isn't symmetric about the axis → shell method (slice parallel to the axis instead).
- Either method can work for most regions — pick whichever avoids solving the curve equation for the other variable.
Steps for full credit:
- Sketch the region and the axis of revolution (2+ points).
- Identify method (disk/washer/shell) and set up radius/height expressions carefully, including any shifts for a non-standard axis.
- Determine correct limits of integration (intersection points of the curves).
- Set up and evaluate the integral.
Decision order when you see an integral:
- Can you simplify algebraically first (expand, split fraction, simplify)?
- Is it a basic/known antiderivative?
- u-substitution — look for a function and its derivative (up to a constant).
- Integration by parts — products of unlike function types (poly × exp, poly × trig, ln, arctan, etc.)
- Trig integral techniques — powers of sin/cos, sec/tan, etc.
- Trig substitution — expressions like √(a²−x²), √(a²+x²), √(x²−a²).
- Partial fractions — rational function P(x)/Q(x), especially after long division if deg P ≥ deg Q.
- Improper integral — infinite limits or discontinuity in the interval → rewrite as a limit.
Let u = inner function, du = derivative dx. Rewrite entirely in u, integrate, substitute back.
- Repeated by-parts (tabular method) for x²eˣ, x²sinx, etc.
- Circular case (eˣsinx, eˣcosx): apply by parts twice, solve algebraically for the integral.
∫sinᵐx cosⁿx dx
- If m (power of sin) is odd: save one sinx, convert rest to cos via sin²x = 1−cos²x, let u = cosx.
- If n (power of cos) is odd: save one cosx, convert rest to sin via cos²x = 1−sin²x, let u = sinx.
- If both even: use power-reduction identities
$$\sin^2x=\frac{1-\cos2x}{2},\qquad \cos^2x=\frac{1+\cos2x}{2}$$
∫tanᵐx secⁿx dx
- If n (power of sec) is even: save sec²x, convert rest with sec²x = 1+tan²x, let u = tanx.
- If m (power of tan) is odd: save secx·tanx, convert rest with tan²x = sec²x−1, let u = secx.
- Know: ∫tanx dx = ln|secx| + C, ∫secx dx = ln|secx+tanx| + C.
| Expression | Substitution | Identity used |
|---|---|---|
| √(a²−x²) | x = a sinθ | 1−sin²θ=cos²θ |
| √(a²+x²) | x = a tanθ | 1+tan²θ=sec²θ |
| √(x²−a²) | x = a secθ | sec²θ−1=tan²θ |
After integrating, draw a right triangle from the substitution to convert θ back to x.
- Make sure deg(numerator) < deg(denominator); if not, do polynomial long division first.
- Factor the denominator completely.
- Set up terms:
- Linear factor (ax+b): term A/(ax+b)
- Repeated linear (ax+b)ⁿ: terms A₁/(ax+b) + A₂/(ax+b)² + … + Aₙ/(ax+b)ⁿ
- Irreducible quadratic (ax²+bx+c): term (Ax+B)/(ax²+bx+c)
- Clear denominators, solve for constants (plug in convenient x-values or match coefficients).
- Integrate each simple piece (logs, arctans).
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Infinite limit:
$\int_a^\infty f,dx = \lim_{t\to\infty}\int_a^t f,dx$ . Converges if the limit is finite. -
Discontinuity at c in [a,b]: split and take one-sided limits:
$\lim_{t\to c^-}\int_a^t f,dx$ , etc. - Both pieces must converge for the whole integral to converge.
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p-integral test:
$\int_1^\infty \frac{1}{x^p}dx$ converges iff p > 1.
A sequence {aₙ} converges to L if
Techniques:
- Treat n as continuous x, use standard limit laws / L'Hôpital's rule (valid since if the continuous limit exists, the sequence limit equals it).
- Squeeze Theorem: if bₙ ≤ aₙ ≤ cₙ and bₙ, cₙ → L, then aₙ → L. Useful for sequences with sin, cos, or (−1)ⁿ factors.
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Key limit:
$\lim_{n\to\infty} r^n = 0$ if |r| < 1; diverges if |r| > 1 (or r = −1, oscillates); = 1 if r = 1. $\lim_{n\to\infty}\left(1+\frac{k}{n}\right)^n = e^k$ - Factorial/exponential growth:
$\lim \frac{n!}{n^n}=0$ , and n! grows faster than any exponential aⁿ. -
Absolute Value Theorem: if
$\lim |a_n| = 0$ then$\lim a_n = 0$ (handles alternating sequences). - Monotonic Sequence Theorem: a bounded, monotonic (always increasing or always decreasing) sequence converges. Useful for recursively defined sequences: show monotonic + bounded ⇒ converges, then take limit of both sides of the recursion to solve for L.
Common divergence indicators: dominant term grows without bound, oscillation without approaching a single value (e.g., (−1)ⁿ, sin(n) without decay), or the sequence is unbounded.
Taylor series of f(x) centered at a:
Maclaurin series = Taylor series centered at a = 0:
Process:
- Compute f(a), f′(a), f″(a), f‴(a), … and find the pattern.
- Write general term
$\dfrac{f^{(n)}(a)}{n!}(x-a)^n$ in summation notation. - Find radius of convergence R using the Ratio Test:
$L=\lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$ ; series converges absolutely when L < 1. - Check the endpoints x = a±R separately (plug in, test with an appropriate series test) to determine the exact interval of convergence (open, half-open, or closed).
Memorize these Maclaurin series (centered at 0), all valid for all x unless noted:
Shortcut: for many problems it is far faster to take a known series and substitute, differentiate, integrate, or multiply by xⁿ rather than compute derivatives directly — check whether the function is a disguised version of one above.
Build from the geometric series:
Common manipulations:
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Substitution: replace x with an expression. E.g.
$\frac{1}{1+x^2}=\frac{1}{1-(-x^2)}=\sum_{n=0}^\infty(-1)^n x^{2n}$ . -
Multiplying by a power of x:
$\frac{x}{1-x}= \sum_{n=0}^\infty x^{n+1}$ . -
Differentiating term-by-term: if
$f(x)=\sum a_n x^n$ , then$f'(x)=\sum n,a_n x^{n-1}$ (same radius of convergence, endpoints re-checked). -
Integrating term-by-term:
$\int f(x),dx = \sum \frac{a_n}{n+1}x^{n+1}+C$ . - Partial fractions first, then apply geometric series to each piece, if the denominator factors.
Finding interval of convergence: apply the Ratio Test to the general term, solve |x−a| < R for R, then test both endpoints individually in the original series (don't assume from the open interval).
Eliminating the parameter t:
- Solve one equation for t, substitute into the other, or
- Use an identity: if x = a cos t, y = b sin t → (x/a)² + (y/b)² = 1 (ellipse), using cos²t + sin²t = 1.
- Watch the domain/range restrictions on t — they restrict the resulting curve (e.g., only part of the ellipse, direction of tracing).
Slope of tangent line (no need to eliminate t):
Second derivative:
Identifying the curve: after eliminating t, recognize standard forms:
- Line, parabola, circle, ellipse, hyperbola (see conic forms in section 9 below).
- Note orientation/direction of tracing from increasing t, and any restricted domain (arrows on the graph).
Graphing: plot a table of (x,y) points for increasing t values, connect in order, and mark the direction of motion with an arrow.
Conversion formulas:
Polar → Cartesian: substitute the above directly into the polar equation, then complete the square if needed to identify the curve.
Cartesian → Polar: substitute x = r cosθ, y = r sinθ into the Cartesian equation and simplify for r (or r²).
Common polar curves to recognize:
| Polar equation | Cartesian form | Curve |
|---|---|---|
| r = a | x² + y² = a² | circle, center origin |
| r = a cosθ | (x − a/2)² + y² = (a/2)² | circle through origin, on x-axis |
| r = a sinθ | x² + (y − a/2)² = (a/2)² | circle through origin, on y-axis |
| θ = c | y = (tan c)x | line through origin |
| r = a + b cosθ (or sinθ), a=b | cardioid | heart shape, cusp at origin |
| r = a + b cosθ, a<b | limaçon with inner loop | |
| r = a + b cosθ, a>b | dimpled/convex limaçon | |
| r² = a² cos2θ | lemniscate | figure-eight |
| r = a cos(nθ) or a sin(nθ) | rose curve | n petals if n odd, 2n petals if n even |
Graphing: Make a table of θ vs r over one full period (usually 0 to 2π, or 0 to π for some roses), plot points, and connect smoothly. Pay attention to:
- Symmetry (about polar axis if r(−θ)=r(θ); about the pole if r(θ+π)=r(θ); about θ=π/2 if r(π−θ)=r(θ)).
- Where r = 0 (curve passes through the pole) and where r is maximum.
Area of a polar region:
Steps:
- Identify the correct bounds α, β where the curve traces the desired region exactly once (find intersection angles by setting r-values equal, or find where r = 0).
-
Area between two curves:
$A = \int_\alpha^\beta \frac{1}{2}\left([r_{outer}]^2-[r_{inner}]^2\right)d\theta$ — always outer minus inner, squared first. - Area of a single inner loop of a limaçon: find the θ-values where r = 0 bounding that loop, integrate only over that range.
- Watch for symmetry to simplify: often easier to integrate over half the region and double it.
- Simplify the integrand with power-reduction identities (
$\cos^2\theta=\frac{1+\cos2\theta}{2}$ , etc.) before integrating.
General strategy: complete the square in x and y to get the standard form, then read off center, a, b (or c), and classify.
- Vertex (h,k). Opens toward the sign of p.
- Focus is p units from vertex along axis of symmetry; directrix is p units on the opposite side.
- Center (h,k). a > b: major axis horizontal (if a² is under the x-term); a < b: major axis vertical.
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$c^2 = a^2 - b^2$ (a is always the larger denominator's square root). - Vertices: (h±a, k) or (h, k±a) along major axis. Foci: (h±c, k) or (h, k±c).
- Center (h,k).
$c^2 = a^2+b^2$ . - Vertices: a units from center along the transverse axis. Foci: c units from center, same axis.
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Asymptotes: through the center with slopes ±b/a (horizontal transverse axis) or ±a/b (vertical transverse axis):
$$y-k=\pm\frac{b}{a}(x-h)\quad\text{or}\quad y-k=\pm\frac{a}{b}(x-h)$$
Graphing tip: plot center first, then vertices, then sketch the guiding box (using a, b) for ellipses/hyperbolas, draw asymptotes as dashed lines for hyperbolas before sketching the curve.
Standard polar conic equations (focus at the pole):
- e = eccentricity, d = distance from focus (pole) to the directrix.
- Denominator with cosθ → directrix is vertical (x = ±d); denominator with sinθ → directrix is horizontal (y = ±d).
- + sign → directrix to the right/above the pole; − sign → directrix to the left/below the pole.
Classify by eccentricity:
| e value | Conic |
|---|---|
| e = 1 | Parabola |
| 0 < e < 1 | Ellipse |
| e > 1 | Hyperbola |
Finding key features:
- Get the equation into the exact form above (divide numerator and denominator so the constant term in the denominator is 1).
- Read off e directly (coefficient of cosθ or sinθ) and solve for d.
- Vertices: plug in the θ-values that point along the axis of symmetry (θ = 0, π for cosθ-form; θ = π/2, 3π/2 for sinθ-form) to get r, then convert (r,θ) to Cartesian points.
- Focus: always at the pole (origin) for this standard form.
- Directrix: the line x = d or y = d (with appropriate sign) as determined above.
- Asymptotes (hyperbola only, e>1): find where r → ∞, i.e., where the denominator = 0; those θ-values give the asymptote directions from the pole (lines through the origin at those angles), or convert the Cartesian form (after multiplying out) to extract slopes.
- Graphing: plot the vertex/vertices found in step 3, note the pole is a focus, and sketch consistent with the eccentricity (nearly circular for small e, elongated for e close to 1, two open branches for e>1).
- Velocity/net change: decide if the problem wants signed net change (just integrate) or total accumulation/distance (integrate |rate|, splitting at sign changes).
- Volumes: graph the region AND the axis of revolution first (guaranteed points); choose disk/washer vs. shell based on which needs fewer variable-swaps; always outer² − inner² for washers.
- Integrals: identify structure first → pick the right technique → simplify before integrating when possible → don't forget +C for indefinite integrals.
- Sequences: find the limit as n→∞; use L'Hôpital, squeeze theorem, or monotonic/bounded argument as needed.
- Series: know your 6 memorized Maclaurin series cold; always find R with Ratio Test, then check both endpoints separately.
- Parametric/Polar/Conics: always convert carefully, complete the square when needed, and double check by identifying key features (vertices, foci, center, asymptotes, directrix) before graphing.
- Polar area: always graph first (4 pts), correctly identify bounds α, β (8 pts) — this is the #1 place points are lost.