The diagram shows a sketch of part of the curves with equations $y = 2 x^{3} + 2 x^{2} - 9 x - 24$ and $y = a x^{3} + b x^{2} + c x + d,$ where a, b, c and d are constants with $a \neq 0$ .

q8a-5-4-further-integration-veryhard-ib-aa-sl

The x-coordinates of the points of intersection of the two curves are p, q and r, where $p < q < r .$ Region S is the region enclosed by the two curves between $x = p$ and $x = q$ , while region T is the region enclosed by the two curves between $x = q$ and $x = r .$

The diagram below shows a sketch of part of the curve with equation $y = 4 x^{3} + 9 x^{2} - 16 x - 21 .$

7_LCsAWI_q9-5-4-further-integration-hard-ib-aa-sl

The curve intersects the x-axis at the points $(p, 0), (q, 0)$ and $(r, 0)$ , and region U is the region enclosed by the curve and the x-axis between $x = p$ and $x = q$ .

Given that the areas of regions S and U are equal, calculate the total area enclosed by the two curves in the first diagram. Be sure to provide a suitable justification for your answer.

Further Integration (DP IB Maths: AA HL)

Exam Questions

1. Number & Algebra

1.1 Number & Algebra Toolkit

1.2 Exponentials & Logs

1.3 Sequences & Series

1.4 Simple Proof & Reasoning

1.5 Further Proof & Reasoning

1.6 Binomial Theorem

1.7 Permutations & Combinations

1.8 Complex Numbers

1.9 Further Complex Numbers

1.10 Systems of Linear Equations

2. Functions

2.1 Linear Functions & Graphs

2.2 Quadratic Functions & Graphs

2.3 Functions Toolkit

2.4 Other Functions & Graphs

2.5 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.7 Polynomial Functions

2.8 Inequalities

2.9 Further Functions & Graphs

3. Geometry & Trigonometry

3.1 Geometry Toolkit

3.2 Geometry of 3D Shapes

3.3 Trigonometry Toolkit

3.4 Trigonometry

3.5 Trigonometric Functions & Graphs

3.6 Trigonometric Equations & Identities

3.7 Inverse & Reciprocal Trig Functions

3.8 Further Trigonometry

3.9 Vector Properties

3.10 Vector Equations of Lines

3.11 Vector Planes

4. Statistics & Probability

4.1 Statistics Toolkit

4.2 Correlation & Regression

4.3 Probability

4.4 Probability Distributions

4.5 Binomial Distribution

4.6 Normal Distribution

4.7 Further Probability Distributions

5. Calculus

5.1 Differentiation

5.2 Further Differentiation

5.3 Integration

5.4 Further Integration

5.5 Optimisation

5.6 Kinematics

5.7 Basic Limits & Continuity

5.8 Advanced Differentiation

5.9 Advanced Integration

5.10 Differential Equations

5.11 MacLaurin Series

5.12 Further Limits (inc l'Hôpital's Rule)

6. Extended Questions

6.1 Extended Questions (Section B, SL)

6.2 Extended Questions (Section B, HL)