Overview
This lecture covers methods for calculating areas, volumes, arc lengths, work, average values, integrating trigonometric functions, improper integrals, sequences, series, convergence tests, Taylor and power series, and parametric and polar coordinates.
Area Between Curves
- The area between two curves y = f(x) and y = g(x) from x = a to x = b is given by the integral ∫ from a to b of (top function − bottom function) dx.
- If f(x) < g(x) on [a, b], switch the order: ∫ from a to b of (g(x) − f(x)) dx to ensure the area is positive.
- To find the limits of integration, set f(x) = g(x) and solve for x to find intersection points.
- The formula only works if the "top" function is always above the "bottom" function on the interval; otherwise, split the interval where the curves cross.
Volumes Using Disks and Washers
- To find the volume of a solid of revolution about the x-axis:
- Disk method: V = ∫ from a to b of π[R(x)]² dx, where R(x) is the distance from the axis to the curve.
- Washer method: V = ∫ from a to b of π[R_outer(x)² − R_inner(x)²] dx, where R_outer and R_inner are the distances from the axis to the outer and inner curves.
- For rotation about the y-axis or a vertical line, use y as the variable and integrate with respect to y.
- The cross-sectional area at each slice is either a disk (solid) or a washer (hollow), depending on the region being rotated.
Volumes by Cross-Sections
- The volume of a solid with known cross-sectional area A(x) perpendicular to the x-axis is V = ∫ from a to b of A(x) dx.
- To find A(x), use the geometry of the cross-section (e.g., for squares, A(x) = [side length]²).
- If cross-sections are perpendicular to the y-axis, integrate with respect to y and use A(y).
Arc Length
- The arc length of a curve y = f(x) from x = a to x = b is L = ∫ from a to b sqrt(1 + [f'(x)]²) dx.
- For parametric curves x = f(t), y = g(t), arc length is L = ∫ from t = a to t = b sqrt([dx/dt]² + [dy/dt]²) dt.
- The formula is derived by approximating the curve with line segments and taking the limit as the segments become infinitely small.
Work
- For a constant force F over a distance d: Work = F × d.
- For a variable force F(x) over [a, b]: Work = ∫ from a to b F(x) dx.
- Units: In metric, work is in joules (newton-meters); in English units, work is in foot-pounds.
- For lifting objects against gravity, use the weight as the force.
Average Value and Mean Value Theorem for Integrals
- The average value of a function f(x) on [a, b] is (1/(b−a)) ∫ from a to b f(x) dx.
- The mean value theorem for integrals states there exists c in [a, b] such that f(c) equals the average value.
- Geometric interpretation: The area under the curve equals the area of a rectangle with height equal to the average value.
Integration by Parts
- Formula: ∫ u dv = uv − ∫ v du.
- Choose u and dv so that ∫ v du is easier to integrate than the original integral.
- Integration by parts is based on the product rule for derivatives.
- Can be used for definite and indefinite integrals.
Trig Identities and Integrals
- Key identities:
- sin²θ + cos²θ = 1 (Pythagorean identity)
- sin(−θ) = −sin θ (odd function)
- cos(−θ) = cos θ (even function)
- Angle sum formulas: sin(A+B) = sin A cos B + cos A sin B; cos(A+B) = cos A cos B − sin A sin B
- Double angle formulas: sin²θ = ½(1 − cos 2θ), cos²θ = ½(1 + cos 2θ)
- Use identities to rewrite integrals with odd or even powers of sine and cosine.
- For odd powers, save one sine or cosine and convert the rest using the Pythagorean identity.
- For even powers, use double angle identities to reduce the power.
Special Trig Integrals & Substitution
- ∫ tan²x dx = tan x − x + C
- ∫ sec x dx = ln|sec x + tan x| + C
- Trig substitution is used for integrals involving square roots:
- For sqrt(a² − x²), use x = a sin θ
- For sqrt(a² + x²), use x = a tan θ
- For sqrt(x² − a²), use x = a sec θ
- After substitution, use right triangle diagrams to convert back to x.
Partial Fractions
- To integrate rational functions, decompose into simpler fractions:
- Example: 3x+2 / (x²+2x−3) = A/(x−1) + B/(x+3)
- Solve for A and B by equating numerators and matching coefficients.
- Works when the denominator factors into distinct linear factors and the numerator's degree is less than the denominator's.
Improper Integrals
- Type I: Integrals over infinite intervals (e.g., ∫₁^∞ 1/x² dx). Defined as a limit as the upper or lower bound approaches infinity.
- Type II: Integrals where the integrand is unbounded on the interval (e.g., ∫₀¹ 1/√x dx). Defined as a limit as the variable approaches the point of discontinuity.
- An improper integral converges if the limit exists and is finite; otherwise, it diverges.
- The comparison test can be used to determine convergence or divergence by comparing to a known integral.
Sequences and Series
- A sequence is an ordered list: a₁, a₂, a₃, ...
- A series is the sum of a sequence: a₁ + a₂ + a₃ + ...
- The sequence of partial sums Sₙ = a₁ + a₂ + ... + aₙ.
- A series converges if the sequence of partial sums approaches a finite limit as n → ∞; otherwise, it diverges.
- Arithmetic sequence: each term increases by a constant difference.
- Geometric sequence: each term is multiplied by a constant ratio.
Convergence Tests
- Geometric series: ∑ arⁿ converges if |r| < 1; sum is a/(1−r).
- p-series: ∑ 1/n^p converges if p > 1; diverges if p ≤ 1.
- Integral test: If f(x) is positive, continuous, and decreasing, then ∑ aₙ and ∫ f(x) dx both converge or both diverge.
- Comparison test: If 0 ≤ aₙ ≤ bₙ and ∑ bₙ converges, then ∑ aₙ converges. If ∑ aₙ diverges and aₙ ≥ bₙ ≥ 0, then ∑ bₙ diverges.
- Limit comparison test: If lim (aₙ/bₙ) = L > 0 and finite, then ∑ aₙ and ∑ bₙ both converge or both diverge.
- Ratio test: lim |aₙ₊₁/aₙ| = L. If L < 1, the series converges absolutely; if L > 1 or L = ∞, it diverges; if L = 1, the test is inconclusive.
- Alternating series test: If terms alternate in sign, decrease in magnitude, and approach zero, the series converges.
- Absolute convergence: If ∑ |aₙ| converges, then ∑ aₙ converges.
- Conditional convergence: If ∑ aₙ converges but ∑ |aₙ| diverges.
Power and Taylor Series
- Power series: ∑ cₙ(x−a)ⁿ; converges for |x−a| < R, where R is the radius of convergence.
- Interval of convergence: The set of x-values for which the series converges; may include or exclude endpoints.
- Taylor series for f(x) at x = a: ∑ [f⁽ⁿ⁾(a)/n!] (x−a)ⁿ.
- Maclaurin series: Taylor series centered at a = 0.
- Taylor's inequality: |Rₙ(x)| ≤ M/(n+1)! |x−a|ⁿ⁺¹, where M bounds the (n+1)th derivative on the interval.
- If all derivatives are bounded on an interval, the Taylor series converges to f(x) on that interval.
- Common Taylor series: eˣ, sin x, cos x, ln(1+x), arctan x, 1/(1−x).
Parametric and Polar Coordinates
- Parametric equations: x = f(t), y = g(t); t is the parameter (often time).
- To eliminate t, solve one equation for t and substitute into the other.
- Slope: dy/dx = (dy/dt)/(dx/dt).
- Arc length: L = ∫ sqrt([dx/dt]² + [dy/dt]²) dt.
- Area under curve: A = ∫ y dx = ∫ y(t) x'(t) dt.
- Polar coordinates: (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis.
- x = r cos θ, y = r sin θ.
- r = √(x² + y²), θ = arctan(y/x).
- To convert between polar and Cartesian, use these relationships.
- Negative r means the point is in the opposite direction from θ.
Key Terms & Definitions
- Riemann sum: An approximation of area or volume using sums of rectangles or other shapes; basis for definite integrals.
- Washer/Disc methods: Techniques for finding volumes of solids of revolution.
- Improper integral: An integral with infinite limits or an unbounded integrand, evaluated using limits.
- Power series: An infinite sum of the form ∑ cₙ(x−a)ⁿ.
- Radius of convergence: The distance from the center a within which a power series converges.
- Taylor series: A power series representation of a function using its derivatives at a single point.
- p-series: A series of the form ∑ 1/n^p.
- Partial fractions: Decomposition of a rational function into simpler fractions for integration.
- Absolute convergence: A series ∑ aₙ is absolutely convergent if ∑ |aₙ| converges.
- Conditional convergence: A series converges, but not absolutely.
- Alternating series: A series whose terms alternate in sign.
Action Items / Next Steps
- Practice using convergence and divergence tests on a variety of series, including geometric, p-series, and alternating series.
- Solve area and volume problems using definite integrals, including those involving disks, washers, and cross-sections.
- Find Taylor and power series representations for common functions and determine their intervals of convergence.
- Review and memorize key trigonometric identities and practice integrating trigonometric functions using substitution and identities.
- Work through problems involving parametric and polar coordinates, including converting between forms and finding slopes, areas, and arc lengths.
- Complete assigned homework and reading to reinforce these concepts and techniques.