Chapter 3: Integration Notes

Sections

1. 3A Areas and the definite integral Mastered

   The definite integral as the area under a curve, evaluating definite integrals with area formulae for triangles, trapezia and circles, and trapping an integral between upper and lower rectangle sums.

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2. 3B The fundamental theorem of calculus Mastered

   Evaluating definite integrals with primitives using the fundamental theorem of calculus, including expanding brackets, splitting fractions and negative powers, and the warning about integrating across an asymptote.

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3. 3C The definite integral and its properties Mastered

   Definite integrals as sums of signed areas, integrals of odd and even functions over symmetric intervals, dissecting the interval, reversing the limits, sums and multiples of functions, and inequalities.

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4. 3D Proving the fundamental theorem Mastered

   The area function A(x), why its derivative is f(x), and how this proves the fundamental theorem of calculus; differentiating an integral with a variable upper limit.

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5. 3E The indefinite integral Mastered

   Indefinite integral notation, the standard integrals of x^n and (ax+b)^n, and finding a function from its derivative and a known point.

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6. 3F Finding areas Mastered

   Finding areas when the curve is below the x-axis or crosses it by splitting at the x-intercepts, and finding areas between a curve and the y-axis.

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7. 3G Areas of compound regions Mastered

   Finding the area between two curves by integrating the upper curve minus the lower curve, finding points of intersection, and splitting compound regions into simpler pieces.

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8. 3H The trapezoidal rule Mastered

   Approximating a definite integral with the trapezoidal rule using equally spaced function values, and using concavity to decide whether the estimate is too big or too small.

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9. 3I The reverse chain rule Mastered

   Integrating functions of the form f'(x)[f(x)]^n with the reverse chain rule, adjusting for a missing constant factor, and evaluating definite integrals this way.

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