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| 1.27 || Proofs of the basic properties of the integral || 84
| 1.27 || Proofs of the basic properties of the integral || 84
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! colspan="3" | PART II: INTUITIVE GEOMETRY
! colspan="3" | 2. SOME APPLICATIONS OF INTEGRATION
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! colspan="3" | Chapter 5: Distance and Angles
| 2.1 || Introduction || 88
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| 1 || Distance || 107
| 2.2 || The area of a region between two graphs expressed as an integral || 88
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| 2 || Angles || 110
| 2.3 || Worked examples || 89
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| 3 || The Pythagoras theorem || 120
| 2.4 || Exercises || 94
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! colspan="3" | Chapter 6: Isometries
| 2.5 || The trigonometric functions || 94
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| 1 || Some standard mappings of the plane || 133
| 2.6 || Integration formulas for the sine and cosine || 94
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| 2 || Isometries || 143
| 2.7 || A geometric description of the sine and cosine functions || 94
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| 3 || Composition of isometries || 150
| 2.8 || Exercises || 94
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| 4 || Inverse of isometries || 155
| 2.9 || Polar coordinates || 94
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| 5 || Characterization of isometries || 163
| 2.10 || The integral for area in polar coordinates || 94
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| 6 || Congruences || 166
| 2.11 || Exercises || 94
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! colspan="3" | Chapter 7: Area and Applications
| 2.12 || Application of integration to the calculation of volume || 94
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| 1 || Area of a disc of radius ''r'' || 173
| 2.13 || Exercises || 94
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| 2 || Circumference of a circle of radius ''r'' || 180
| 2.14 || Application of integration to the calculation of work || 94
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! colspan="3" | PART III: COORDINATE GEOMETRY
| 2.15 || Exercises || 94
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! colspan="3" | Chapter 8: Coordinates and Geometry
| 2.16 || Average value of a function || 94
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| 2.17 || Exercises || 94
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| 2.18 || The integral as a function of the upper limit. Indefinite integrals || 94
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| 2.19 || Exercises || 94
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! colspan="3" | 3. CONTINUOUS FUNCTIONS
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| 3.1 || Informal description of continuity || 126
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| 3.2 || The definition of the limit of a function || 127
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| 3.3 || The definition of continuity of a function || 130
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| 3.4 || The basic limit theorems. More examples of continuous functions || 131
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| 3.5 || Proofs of the basic limit theorems || 135
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| 3.6 || Exercises || 138
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| 3.7 || Composite functions and continuity || 140
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| 3.8 || Exercises || 142
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| 3.9 || Bolzano's theorem for continuous functions || 142
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| 3.10 || The intermediate-value theorem for continuous functions || 144
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| 3.11 || Exercises || 145
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| 3.12 || The process of inversion || 146
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| 3.13 || Properties of functions preserved by inversion || 147
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| 3.14 || Inverses of piecewise monotonic functions || 148
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| 3.15 || Exercises || 149
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| 3.16 || The extreme-value theorem for continuous functions || 150
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| 3.17 || The small-span theorem for continuous functions (uniform continuity) || 152
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| 3.18 || The integrability theorem for continuous functions || 152
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| 3.19 || Mean-value theorems for integrals of continuous functions || 154
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| 3.20 || Exercises || 155
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! colspan="3" | 4. DIFFERENTIAL CALCULUS
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| 4.1 || Historical introduction || 156
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| 4.2 || A problem involving velocity || 157
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| 4.3 || The derivative of a function || 159
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| 4.4 || Examples of derivatives || 161
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| 4.5 || The algebra of derivatives || 164
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| 4.6 || Exercises || 167
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| 4.7 || Geometric interpretation of the derivative as a slope || 169
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| 4.8 || Other notations for derivatives || 171
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| 4.9 || Exercises || 173
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| 4.10 || The chain rule for differentiating composite functions || 174
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| 4.11 || Applications of the chain rule. Related rates and implicit differentiation || 176
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| 4.12 || Exercises || 179
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| 4.13 || Applications of the differentiation to extreme values of cuntions|| 181
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| 4.14 || The mean-value theorem for derivatives || 183
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| 4.15 || Exercises || 186
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| 4.16 || Applications of the mean-value theorem to geometric properties of functions || 187
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| 4.17 || Second-derivative test for extrema || 188
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| 4.18 || Curve sketching || 189
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| 4.19 || Exercises || 191
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| 4.20 || Worked examples of extremum problems || 191
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| 4.21 || Exercises || 194
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| 4.22 || Partial derivatives || 196
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| 4.23 || Exercises || 201
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! colspan="3" | 5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION
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| 1 || Coordinate systems || 191
| 1 || Coordinate systems || 191
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