Calculus II: Comprehensive Study Notes on Integration Techniques and Applications

Introduction to Integration and Area Under Velocity Graphs

  • The Velocity-Distance Relationship:

    • The area under the velocity graph, $v(t)$, gives the distance traveled.

    • Formula: Distance=Speed×Time\text{Distance} = \text{Speed} \times \text{Time}.

    • Geometric representation: The area of the rectangle formed by the velocity function $f'(x)$.

  • Distance Approximation Example:

    • Calculation: (80×3.60)+(20×1.8)+(40×1.0)=144 km/h(80 \times 3.60) + (20 \times 1.8) + (40 \times 1.0) = 144 \, \text{km/h}.

    • Another calculation provided for a velocity graph: (12.5+5)+10=17.5+10=27.5(12.5 + 5) + 10 = 17.5 + 10 = 27.5.

Riemann Sums and Approximations

  • Definition of Riemann Sums:

    • Approximations are used to find the area under a curve $f(x)$ on an interval $[a, b]$ by dividing it into $n$ partitions of width Δx=b−an\Delta x = \frac{b - a}{n}.

  • Summation Methods:

    1. Left Endpoint Approximation ($L_n$): Ln=∑i=0n−1f(xi)ΔxL_n = \sum_{i=0}^{n-1} f(x_i) \Delta x.

    2. Right Endpoint Approximation ($R_n$): Rn=∑i=1nf(xi)ΔxR_n = \sum_{i=1}^n f(x_i) \Delta x.

    3. Midpoint Approximation ($M_n$): Mn=∑k=1nf(xk∗)ΔxM_n = \sum_{k=1}^n f(x_k^*) \Delta x, where xk∗=xk−1+xk2x_k^* = \frac{x_{k-1} + x_k}{2}.

  • Numerical Example ($x^2$ function on $[0, 8]$ with 4 partitions):

    • Δx=8−04=2\Delta x = \frac{8 - 0}{4} = 2.

    • $L_4$ Calculation: (02+22+42+62)×2=(0+4+16+36)×2=112(0^2 + 2^2 + 4^2 + 6^2) \times 2 = (0 + 4 + 16 + 36) \times 2 = 112.

    • $R_4$ Calculation: (22+42+62+82)×2=120×2=240(2^2 + 4^2 + 6^2 + 8^2) \times 2 = 120 \times 2 = 240.

    • $M_4$ Calculation: (12+32+52+72)×2=(1+9+25+49)×2=168(1^2 + 3^2 + 5^2 + 7^2) \times 2 = (1 + 9 + 25 + 49) \times 2 = 168.

    • Inequality relationship established: L_n < M_n < R_n.

  • Example with 8 partitions:

    • Δx=4−08=0.5\Delta x = \frac{4 - 0}{8} = 0.5.

  • The Best Approximation:

    • Occurs when Δx→0\Delta x \rightarrow 0 (or $n \rightarrow \infty$).

    • Definition: lim⁡Δx→0∑i=1nf(xi)Δx=∫abf(x) dx\lim_{\Delta x \rightarrow 0} \sum_{i=1}^n f(x_i) \Delta x = \int_a^b f(x) \, dx.

Definition of the Riemann Integral

  • General Limit:

    • ∫abf(x) dx=lim⁡n→∞Ln=lim⁡n→∞Mn=lim⁡n→∞Rn\int_a^b f(x) \, dx = \lim_{n \rightarrow \infty} L_n = \lim_{n \rightarrow \infty} M_n = \lim_{n \rightarrow \infty} R_n.

    • If this limit exists, we call it the Riemann Integral of $f(x)$.

    • Parameters:

      • Δx=b−an\Delta x = \frac{b - a}{n}.

      • x0=ax_0 = a.

      • xn=bx_n = b.

  • Visual Interpretation of Velocity Graph:

    • Example: At context $T=3.5$ and $V=60$.

    • D=V×T=3.5×60=210 kmD = V \times T = 3.5 \times 60 = 210 \, \text{km}.

    • Area of a triangle component: A=12×b×h=12×3.5×70=122.5A = \frac{1}{2} \times b \times h = \frac{1}{2} \times 3.5 \times 70 = 122.5.

    • Conclusion: Displacement is the area under the graph of the velocity function.

Properties and Techniques of Integration

  • Constant Multiple Rule:

    • ∫k f(x) dx=k∫f(x) dx\int k \, f(x) \, dx = k \int f(x) \, dx.

  • Sum and Difference Rule:

    • ∫(f±g)(x) dx=∫f(x) dx±∫g(x) dx\int (f \pm g)(x) \, dx = \int f(x) \, dx \pm \int g(x) \, dx.

  • Interval Reversal:

    • ∫abf(x) dx=−∫baf(x) dx\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx.

  • Average Value of a Function:

    • If $f$ is integrable on $[a, b]$, the average value is: favg=1b−a∫abf(x) dxf_{\text{avg}} = \frac{1}{b - a} \int_a^b f(x) \, dx.

    • Example: For $f(x) = 5x^2$ on $[0, 4]$:

      • ∫045x2 dx=5x33∣04=53(64)=3203\int_0^4 5x^2 \, dx = \left. \frac{5x^3}{3} \right|_0^4 = \frac{5}{3}(64) = \frac{320}{3}.

      • favg=14−0×3203=14×3203=803f_{\text{avg}} = \frac{1}{4 - 0} \times \frac{320}{3} = \frac{1}{4} \times \frac{320}{3} = \frac{80}{3}.

The Fundamental Theorems of Calculus (FTC)

  • Antiderivatives:

    • If $f$ is integrable in $[a, b]$, there exists a function $F$ such that $F'(x) = f(x)$. $F$ is the antiderivative of $f$.

  • FTC Part I:

    • If $f$ is continuous in $[a, b]$, then A(x)=∫kxf(t) dtA(x) = \int_k^x f(t) \, dt is a unique antiderivative that satisfies $A(k) = 0$ for $k \in \mathbb{R}$.

  • FTC Part II:

    • If $f$ is continuous and integrable on $[a, b]$ and $F$ is the antiderivative of $f$, then:

    • ∫abf(x) dx=F(b)−F(a)\int_a^b f(x) \, dx = F(b) - F(a).

    • General form: F(b)=F(a)+∫abf(t) dtF(b) = F(a) + \int_a^b f(t) \, dt.

Common Integration Rules

  • Power Rule:

    • ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C (for n≠−1n \neq -1).

  • Exponential and Logarithmic Rules:

    • ∫ex dx=ex+C\int e^x \, dx = e^x + C.

    • ∫1x dx=ln⁡∣x∣+C\int \frac{1}{x} \, dx = \ln|x| + C.

    • ddx(5x)=5xln⁡(5)\frac{d}{dx}(5^x) = 5^x \ln(5), implying ∫5x dx=5xln⁡(5)+C\int 5^x \, dx = \frac{5^x}{\ln(5)} + C.

  • Trigonometric Rules:

    • ∫sin⁡(x) dx=−cos⁡(x)+C\int \sin(x) \, dx = -\cos(x) + C.

    • ∫cos⁡(x) dx=sin⁡(x)+C\int \cos(x) \, dx = \sin(x) + C.

Integration by Substitution (u-substitution)

  • Concept: Make a substitution that simplifies the integrand.

  • Procedure Examples:

    1. Exponentials: ∫e7x dx\int e^{7x} \, dx. Let u=7xu = 7x, dx=17dudx = \frac{1}{7} du. Result: 17e7x+C\frac{1}{7} e^{7x} + C.

    2. Polynomials: ∫(3x−2)4 dx\int (3x - 2)^4 \, dx. Let u=3x−2u = 3x - 2, du=3 dxdu = 3 \, dx. Result: 13(3x−2)55=115(3x−2)5+C\frac{1}{3} \frac{(3x - 2)^5}{5} = \frac{1}{15}(3x - 2)^5 + C.

    3. Composites: ∫2x(3x2−2)3 dx\int 2x(3x^2 - 2)^3 \, dx. Let u=3x2−2u = 3x^2 - 2, du=6x dxdu = 6x \, dx, so x dx=16dux \, dx = \frac{1}{6} du. Result: 2∫u316du=13u44=112(3x2−2)4+C2 \int u^3 \frac{1}{6} du = \frac{1}{3} \frac{u^4}{4} = \frac{1}{12}(3x^2 - 2)^4 + C.

    4. Trig functions: ∫sin⁡(7x3−6)x2 dx\int \sin(7x^3 - 6) x^2 \, dx. Let u=7x3−6u = 7x^3 - 6, du=21x2 dxdu = 21x^2 \, dx. Result: 121∫sin⁡(u) du=−121cos⁡(7x3−6)+C\frac{1}{21} \int \sin(u) \, du = -\frac{1}{21} \cos(7x^3 - 6) + C.

Integration by Parts

  • Formula: Derived from the product rule: ∫u dv=uv−∫v du\int u \, dv = uv - ∫ v \, du.

  • Standard Examples:

    1. Product of polynomial and exponential: ∫xex dx\int x e^x \, dx. Let u=x,dv=ex dxu = x, dv = e^x \, dx. Result: xex−ex+Cx e^x - e^x + C.

    2. Logarithms: ∫ln⁡(x) dx\int \ln(x) \, dx. Let u=​ln⁡(x),dv=dxu = ​\ln(x), dv = dx. Result: xln⁡(x)−x+Cx \ln(x) - x + C.

    3. Cyclic Integration: ∫exsin⁡(x) dx\int e^x \sin(x) \, dx. After two applications of parts, result: 12ex(sin⁡(x)−cos⁡(x))+C\frac{1}{2} e^x (\sin(x) - \cos(x)) + C.

    4. Complex Power/Trig: ∫x2sin⁡(x) dx\int x^2 \sin(x) \, dx. Let u=x2,dv=sin⁡(x) dxu = x^2, dv = \sin(x) \, dx. Result: −x2cos⁡(x)+2xsin⁡(x)+2cos⁡(x)+C-x^2 \cos(x) + 2x \sin(x) + 2 \cos(x) + C.

Integration by Partial Fractions

  • Process: Decompose rational functions into simpler fractions.

  • Example 1: ∫6x+5x2−3x+2 dx\int \frac{6x + 5}{x^2 - 3x + 2} \, dx.

    • Factor denominator: (x−1)(x−2)(x - 1)(x - 2).

    • Decomposition: Ax−1+Bx−2\frac{A}{x - 1} + \frac{B}{x - 2}.

    • Solve coefficients: A+B=6A + B = 6 and −2A−B=5-2A - B = 5. Result: A=−11,B=17A = -11, B = 17.

    • Result: −11ln⁡∣x−1∣+17ln⁡∣x−2∣+C-11 \ln|x - 1| + 17 \ln|x - 2| + C.

  • Example 2: ∫x2+1x3−x2 dx\int \frac{x^2 + 1}{x^3 - x^2} \, dx.

    • Decomposition: ∫(Ax+Bx2+Cx−1) dx\int (\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x - 1}) \, dx.

    • Calculated coefficients: A=−1,B=−1,C=2A = -1, B = -1, C = 2.

    • Result: −ln⁡∣x∣+1x+2ln⁡∣x−1∣+C-\ln|x| + \frac{1}{x} + 2 \ln|x - 1| + C.

Trigonometric Substitution and Identities

  • Key Identities:

    • sin⁡2(x)+cos⁡2(x)=1\sin^2(x) + \cos^2(x) = 1

    • tan⁡2(x)+1=sec⁡2(x)\tan^2(x) + 1 = \sec^2(x)

    • cos⁡(2a)=cos⁡2(a)−sin⁡2(a)=2cos⁡2(a)−1\cos(2a) = \cos^2(a) - \sin^2(a) = 2 \cos^2(a) - 1

  • Standard Substitutions:

    • For a2−x2\sqrt{a^2 - x^2}, let x=asin⁡(θ)x = a \sin(\theta).

    • For a2+x2\sqrt{a^2 + x^2}, let x=atan⁡(θ)x = a \tan(\theta).

    • For x2−a2\sqrt{x^2 - a^2}, let x=asec⁡(θ)x = a \sec(\theta).

  • Reduction Formulas:

    • ∫sin⁡n(x) dx=−1nsin⁡n−1(x)cos⁡(x)+n−1n∫sin⁡n−2(x) dx\int \sin^n(x) \, dx = -\frac{1}{n} \sin^{n-1}(x) \cos(x) + \frac{n-1}{n} \int \sin^{n-2}(x) \, dx.

    • ∫cos⁡n(x) dx=1ncos⁡n−1(x)sin⁡(x)+n−1n∫cos⁡n−2(x) dx\int \cos^n(x) \, dx = \frac{1}{n} \cos^{n-1}(x) \sin(x) + \frac{n-1}{n} \int \cos^{n-2}(x) \, dx.

  • Special Cases for Trigonometric Powers:

    • ∫sin⁡m(x)cos⁡n(x) dx\int \sin^m(x) \cos^n(x) \, dx:

      • If $m$ is odd, let u=cos⁡(x)u = \cos(x).

      • If $n$ is odd, let u=sin⁡(x)u = \sin(x).

      • If both are even, use double angle identities.

Geometric Applications of the Integral

  • Area Between Two Curves:

    • Area A=∫ab(f(x)−g(x)) dxA = \int_a^b (f(x) - g(x)) \, dx, where $f(x) \geq g(x)$.

    • Example: Curve f(x)=x+2f(x) = x + 2 and g(x)=x2g(x) = x^2.

      • Intersection points: x2−x−2=0→(x−2)(x+1)=0→x=2,−1x^2 - x - 2 = 0 \rightarrow (x - 2)(x + 1) = 0 \rightarrow x = 2, -1.

      • Area: ∫−12(x+2−x2) dx=[12x2+2x−13x3]−12=92 units2\int_{-1}^2 (x + 2 - x^2) \, dx = [\frac{1}{2} x^2 + 2x - \frac{1}{3} x^3]_{-1}^2 = \frac{9}{2} \, \text{units}^2.

  • Length of a Curve (Arc Length):

    • Formula: L=∫ab1+(f′(x))2 dxL = \int_a^b \sqrt{1 + (f'(x))^2} \, dx.

    • Example: y=x3/2y = x^{3/2} for 0≤x≤10 \leq x \leq 1. f′(x)=32x1/2f'(x) = \frac{3}{2} x^{1/2}.

    • L=∫011+94x dx=827[(1+94)3/2−1]=827[(134)3/2−1]L = \int_0^1 \sqrt{1 + \frac{9}{4} x} \, dx = \frac{8}{27} [ (1 + \frac{9}{4})^{3/2} - 1] = \frac{8}{27} [(\frac{13}{4})^{3/2} - 1].

  • Surface Area of Revolution:

    • Rotation about x-axis: S=∫ab2πf(x)1+(f′(x))2 dxS = \int_a^b 2 π f(x) \sqrt{1 + (f'(x))^2} \, dx.

    • Rotation about y-axis: S=∫cd2πf(y)1+(f′(y))2 dyS = \int_c^d 2 π f(y) \sqrt{1 + (f'(y))^2} \, dy.

Calculating Volumes of Solids

  • Disk Method:

    • V=∫abπ[f(x)]2 dxV = ∫_a^b π [f(x)]^2 \, dx.

    • Example: $y = ​\sqrt{x}$ rotated about x-axis from 0 to 9: V=∫09πx dx=π[x22]09=40.5πV = ∫_0^9 π x \, dx = π [\frac{x^2}{2}]_0^9 = 40.5π.

  • Washer Method:

    • Used for solids with holes. Formula: V=π∫ab(R2−r2) dxV = π \int_a^b (R^2 - r^2) \, dx.

    • Example: Area between y=4−x2y = 4 - x^2 and y=x+2y = x + 2.

  • Cylindrical Shell Method:

    • Formula: V=∫ab2πxf(x) dxV = \int_a^b 2 π x f(x) \, dx.

    • Example: Volume enclosed by $y = x$ and $y = x^2$ rotated about y-axis: V=∫012πx(x−x2) dx=2π[x33−x44]01=2π(112)=π6V = \int_0^1 2 π x(x - x^2) \, dx = 2 \pi [\frac{x^3}{3} - \frac{x^4}{4}]_0^1 = 2 \pi (\frac{1}{12}) = \frac{π}{6}.

  • Volume of a Sphere Derivation:

    • Equation: x2+y2=r2→y=r2−x2x^2 + y^2 = r^2 \rightarrow y = \sqrt{r^2 - x^2}.

    • V=∫−rrπ(r2−x2)2 dx=π[r2x−13x3]−rr=43πr3V = \int_{-r}^r \pi (\sqrt{r^2 - x^2})^2 \, dx = \pi [r^2 x - \frac{1}{3} x^3]_{-r}^r = \frac{4}{3} \pi r^3.

Improper Integrals

  • Types:

    1. Infinite limits of integration (Type I).

    2. Integrand has a point of discontinuity within the limits (Type II).

  • Convergence and Divergence:

    • If the limit exists and is finite, the integral converges.

    • If the limit does not exist or is infinite, the integral diverges.

  • Example: ∫1∞1x3 dx\int_1^{\infty} \frac{1}{x^3} \, dx.

    • Calculation: lim⁡k→∞∫1kx−3 dx=lim⁡k→∞[−12x2]1k=lim⁡k→∞(−12k2+12)=12\lim_{k \rightarrow \infty} \int_1^k x^{-3} \, dx = \lim_{k → ∞} [-\frac{1}{2x^2}]_1^k = \lim_{k → ∞} (-\frac{1}{2k^2} + \frac{1}{2}) = \frac{1}{2}.

    • The integral converges to 0.50.5.