Integral calculus
Integral calculus for IB Analysis and Approaches HL: antiderivatives, substitution, parts, differential equations, Euler's method and definite integrals.
The integral reverses the derivative: given the rate of change of a quantity, it recovers the quantity. In this topic you learn to calculate antiderivatives with the Higher Level techniques (substitution, parts and partial fractions), to solve first-order differential equations (separable, homogeneous and linear) and to approximate their solutions with Euler's method. Then you move on to the definite integral: the fundamental theorem of calculus, Barrow's rule, the area between curves, the volume of a solid of revolution and improper integrals. It appears in all Higher Level papers, with and without a calculator.
Recommended order
Follow the sections in order: antiderivatives, substitution, parts, rational functions, differential equations (separable, homogeneous and linear), Euler's method, definite integral, fundamental theorem and Barrow's rule, areas, volumes of revolution and improper integrals. At the end you have IB exam questions grouped by type. Before starting, review the derivatives from topic 10: integrating is mostly about recognising which derivative you have “undone”.
Differential calculus (topic 10) and functions (topic 9).
Sections in this topic
Antiderivatives and the indefinite integral
Antiderivatives and the constant C, properties of the indefinite integral and a table of standard integrals, including f(ax+b). IB Analysis and Approaches HL.
3 worked examples Section 2Integration by substitution
Substitution in indefinite and definite integrals: choosing u, changing the limits and using the substitution given in exams. IB Analysis and Approaches HL.
3 worked examples Section 3Integration by parts
Integration by parts: the formula, choosing u and dv with ILATE, and cyclic integrals such as e^x sin x. IB Analysis and Approaches HL.
3 worked examples Section 4Integrating rational functions
Integrating rational functions with division and partial fractions: distinct, repeated and quadratic factors. IB Analysis and Approaches HL.
3 worked examples Section 5Separable differential equations
Separable differential equations: general and particular solutions, initial conditions and context problems. IB Analysis and Approaches HL.
3 worked examples Section 6Homogeneous differential equations
Homogeneous differential equations: recognising dy/dx = F(y/x) and solving it with y = vx, which makes it separable. IB Analysis and Approaches HL.
3 worked examples Section 7Linear first-order differential equations
Linear first-order differential equations: integrating factor, general and particular solutions, with worked examples. IB Analysis and Approaches HL.
3 worked examples Section 8Euler's method
Euler's method for approximating differential equation solutions: formula, tables, error, step size and slope fields. IB Analysis and Approaches HL.
3 worked examples Section 9Definite integral and signed area under a curve
Definite integral and signed area: lower and upper Riemann sums, the limit, geometric interpretation and worked examples. IB Analysis and Approaches HL.
3 worked examples Section 10Fundamental theorem of calculus and Barrow's rule
Fundamental theorem of calculus, Barrow's rule, properties of the definite integral, symmetry of even and odd functions and mean value, with IB-style examples.
3 worked examples Section 11Calculating areas with integrals
How to find the area between a curve and the x-axis or between two curves, with sign changes and intersection points, and the distance travelled from velocity.
3 worked examples Section 12Volumes of solids of revolution
Volume of solids of revolution with integrals: rotation about the x- or y-axis, disc and washer methods, with worked examples for IB Analysis and Approaches HL.
3 worked examples Section 13Improper integrals
Integrals with an infinite limit or an unbounded function: definition with limits, convergence, divergence, the power test and infinite areas.
3 worked examples End of topicIB exam questions on integral calculus
IB exam questions on integral calculus: antiderivatives, differential equations, Euler's method, areas, volumes and improper integrals, with solutions.
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