Advanced Calculus Review: Integration, Series, and Polar Coordinates

What are the Pythagorean Identities?

sin² x = 1 - cos² x, cos² x = 1 - sin² x, sec² x = 1 + tan² x, tan² x = sec² x - 1, csc² x = 1 + cot² x, cot² x = csc² x - 1.

What are the Double Angle Formulas for cosine and sine?

cos² x = 1/2(1 + cos 2x), sin² x = 1/2(1 - cos 2x).

How do you integrate products of sines and cosines when at least one exponent is odd?

Break the odd exponent down into an even and a single power, using the single power as 'du'.

What should you do if both exponents of sines and cosines are odd?

Break down the smaller exponent into an even and a single power.

What is the approach for integrating products of tangents and secants?

Isolate either sec x tan x or sec² x to use as 'du', and rewrite everything else using Pythagorean Identities.

What is the substitution for √(a² - x²)?

Use x = a sin θ.

What is the substitution for √(x² - a²)?

Use x = a sec θ.

What is the substitution for √(x² + a²)?

Use x = a tan θ.

What must you remember when substituting in integrals?

dx ≠ dθ; take the derivative of the substitution to properly substitute.

What is the first step when integrating a rational function?

Check if substitution or integration by parts can be applied.

What should you do if the degree of the numerator is greater than or equal to the degree of the denominator?

Perform long division first.

What is included in the partial fraction decomposition for a distinct term (x - a)?

A/(x - a).

What is included in the partial fraction decomposition for a repeated linear term (x - b)²?

A/(x - b) + B/(x - b)².

What is included in the partial fraction decomposition for an irreducible quadratic term?

Ax + B as its numerator.

What is the formula for the Midpoint Rule in numerical integration?

Mn = ∆x Σ f(ci), where ci is the midpoint of the ith subinterval.

What is the formula for the Trapezoid Rule?

Tn = (1/2) · ∆x [f(a) + 2Σ f(xi) + f(b)].

What is the formula for Simpson's Rule?

Sn = (1/3) · ∆x [f(a) + 4Σ f(xi) + 2Σ f(xi) + f(b)], where n must be even.

What is the error formula for the Midpoint Rule?

|EMn| ≤ K(b - a)³ / (24n²).

What is the error formula for the Trapezoid Rule?

|ETn| ≤ K(b - a)³ / (12n²).

What is the error formula for Simpson's Rule?

|ESn| ≤ L(b - a)⁵ / (180n⁴).

What defines an improper integral?

An integral where the Fundamental Theorem of Calculus does not apply, often due to infinite limits or discontinuities.

How does an improper integral converge?

If the limit associated with it is finite.

How does an improper integral diverge?

If the limit associated with it is infinite.

What is the limit definition for an improper integral with an infinite upper limit?

∫[a, ∞) f(x) dx = lim (t→∞) ∫[a, t] f(x) dx.

What is the limit definition for an improper integral with an infinite lower limit?

∫(-∞, b] f(x) dx = lim (t→-∞) ∫[t, b] f(x) dx.

What is the limit definition for an infinite integral from a to infinity?

If f is continuous on [a, ∞), then ∫[a, ∞) f(x) dx = lim (t→∞) ∫[a, t] f(x) dx.

How do you evaluate an integral from negative infinity to b?

If f is continuous on (−∞, b], then ∫[−∞, b] f(x) dx = lim (t→−∞) ∫[t, b] f(x) dx.

What is the condition for the convergence of an integral over the entire real line?

If f is continuous on (−∞, ∞), then ∫[−∞, ∞) f(x) dx = ∫[−∞, c] f(x) dx + ∫[c, ∞) f(x) dx, and both integrals must converge.

What is the limit definition for a discontinuous integral at the lower limit?

If f is continuous on (a, b] but discontinuous at x = a, then ∫[a, b] f(x) dx = lim (t→a+) ∫[t, b] f(x) dx.

What is the limit definition for a discontinuous integral at the upper limit?

If f is continuous on [a, b) but discontinuous at x = b, then ∫[a, b] f(x) dx = lim (t→b−) ∫[a, t] f(x) dx.

What is the formula to find the area enclosed by a polar curve?

To find the area A enclosed by a polar curve, use A = 1/2 ∫[α, β] [f(θ)]² dθ over the interval where the curve is traced exactly once.

How do you convert polar coordinates to Cartesian coordinates?

Use the relationships x = r cos(θ) and y = r sin(θ).

What is the condition for a sequence to converge?

A sequence {an} converges if lim (n→∞) an = L, where L is finite.

What does it mean for a sequence to be monotonic?

A sequence is monotonic if it is either increasing (ak+1 ≥ ak) or decreasing (ak+1 ≤ ak).

What is the Squeeze Theorem in relation to sequences?

If f(n) = an for all n and lim (x→∞) f(x) = L, then lim (n→∞) an = L as well.

What is the condition for a geometric series to converge?

A geometric series converges if |r| < 1, where r is the common ratio.

What is the formula for the sum of a converging geometric series?

The sum S of a converging geometric series is S = a / (1 - r), where a is the first term.

What is a telescoping series?

A telescoping series is one where each term can be expressed as a difference, allowing for cancellation of terms.

What is the test for divergence for a series?

If lim (n→∞) an ≠ 0, then the series ∑ an diverges.

What is the Comparison Theorem for integrals?

If f and g are continuous on [a, ∞) and f(x) ≥ g(x) ≥ 0, then if ∫[a, ∞) f(x) dx converges, so does ∫[a, ∞) g(x) dx.

What happens if one of the series in a pair converges and the other diverges?

If ∑ an converges and ∑ bn diverges, there is no conclusion about the convergence of the combined series.

What is the relationship between bounded monotonic sequences and convergence?

Every bounded monotonic sequence converges.

How do you determine the slope of a tangent line to a polar curve?

The slope is given by dy/dx = (dr/dθ cos(θ) - r sin(θ)) / (dr/dθ cos(θ) - r sin(θ)).

What should you be careful about when taking limits in calculus?

Be sure to evaluate limits 'from the inside out' and consider using L'Hôpital's Rule if necessary.

What is the significance of changing variables in integration?

When changing variables using substitution, ensure to change the bounds accordingly.

How can you find the points of intersection for two polar equations?

Set the equations equal to each other to find θ, then substitute to find r.

What do you need to do to find the area between two polar curves?

First, find the intersection points, then use the area formula A = 1/2 ∫[α, β] [f(θ)]² dθ, adjusting signs as needed.