AP Calc Ab Integration Notes

  1. Fundamental Theorem of Calculus (FTC)

    • Part 1: Establishes the connection between differentiation and integration.

      • If f is continuous on [a, b] and F is an antiderivative of f on [a, b], then:[ \int_a^b f(x) \ dx = F(b) - F(a) ]

    • Part 2: Describes how differentiation works with integrals.

      • If F(x) = [ \int_a^x f(t) \ dt ], then:[ F'(x) = f(x) ]

    • Interpretation: This shows that differentiation "undoes" integration, allowing evaluation of integrals using antiderivatives.

  2. U-Substitution (U-Sub)

    • Choose u: Select u = g(x) where g(x) is a suitable function to simplify the integral.

    • Compute du: Compute the differential: [ du = g'(x) \ dx ].

    • Rewrite Integral: Substitute u into the integral:[ \int f(g(x)) g'(x) \ dx = \int f(u) \ du ]

    • Integrate: Solve the integral in terms of u.

    • Substitute Back: Replace u back in terms of x after integrating to get the final result.

  3. Complex Integrals

    • Identify the Function: Determine the function of complex numbers to integrate.

    • Identifying Contour: Choose a closed contour C in the complex plane for integration.

    • Residue Theorem: Use the theorem when evaluating around poles. If z has a pole, the integral around the contour can be expressed as:[ \int_C f(z) \ dz = 2 \pi i \times \text{(Sum of residues inside C)} ]

    • Interpretation: This simplifies evaluation of integrals of complex functions.

  4. Inverse Trigonometric Functions

    • Identify the Integral: Recognize forms for integrals involving inverse trigonometric functions:

      • For example:[ \int \frac{1}{\sqrt{1-x^2}} \ dx = \sin^{-1}(x) + C ]

    • Evaluate Other Forms: Learn and evaluate other forms:

      • [ \int \frac{1}{1+x^2} \ dx = \tan^{-1}(x) + C ]

      • [ \int \frac{1}{|x| \sqrt{x^2-1}} \ dx = \sec^{-1}(|x|) + C ]

  5. Definite Integration

    • Set Up the Integral: Write the definite integral over [a, b]:[ A = \int_a^b f(x) \ dx ]

    • Find Antiderivative: Calculate the antiderivative F(x) of f(x).

    • Apply Limits: Compute the definite integral using limits:[ A = F(b) - F(a) ]

    • Interpretation: The result A represents total accumulated value under the curve from x = a to b.

  6. FTC Derivatives

    • Choose Function: Let F(x) = [ \int_a^x f(t) \ dt ].

    • Apply FTC: By the theorem:[ F'(x) = f(x) ]

    • Interpretation: This shows that differentiating the definite integral of f(t) from a to x yields f(x).

  7. Finding Area Under a Curve

    • Identify the Function: Determine f(x) and the interval from a to b.

    • Set Up the Integral: Write the integral for area under the curve:[ A = \int_a^b f(x) \ dx ]

    • Find Antiderivative: Calculate F(x), the antiderivative of f(x).

    • Apply Limits: Compute the integral using limits:[ A = F(b) - F(a) ]

    • Interpretation: Result A represents area under the curve from x = a to x = b.

  8. Finding Area Between a Curve and the X-Axis

    • Identify Function: Determine f(x) and the interval from a to b.

    • Set Up the Integral: Write integral for area:[ A = \int_a^b |f(x)| \ dx ]

    • Determine Crossings: Check for crossings of the x-axis.

    • Separate Segments: If crossings occur, divide integral into segments where f(x) > 0 and f(x) < 0.

    • Compute Each Integral: Evaluate integrals separately and sum absolute areas to obtain total area.

  9. Finding Area Between Two Curves

    • Identify Both Functions: Determine upper curve f(x) and lower curve g(x) over interval [a, b].

    • Set Up the Integral: Write area formula:[ A = \int_a^b (f(x) - g(x)) \ dx ]

    • Find Points of Intersection: Identify intersection points to determine limits if needed.

    • Evaluate Integral: Compute the integral to find area between the curves.

    • Interpretation: Result A represents the area between two curves from x = a to x = b.