YEAR 1 INTEGRATION Definite Integrals Notes Examples Exercise

YEAR 1 -INTEGRATION Definite Integrals: Notes | Examples | Exercise Area Under the x-axis : Notes | Examples | Exercise Area Between a Curve and a Line: Notes | Examples | Exercise Area Between two curves: Notes | Examples | Exercise Exploring Definite Integrals Further: Exercise Mixed Integration Practice: Questions 25/05/2021 1

25/05/2021 differentiate Integration is the reverse of differentiation integrate Divide by the new power Increase the power by 1 2

INTEGRATION PRACTICE 25/05/2021 You have already completed a number of practice problems on integration in the flipped learning homework tasks/. Use the questions on the next two pages to refresh your memory and improve your fluency. EXERCISE 14 B | QUESTIONS: P 258 -259 | ANSWERS: P 566 Q 2: (a)(i)(c)(ii) Q 3: (a)(ii) (b)(ii) Q 4 -13 (you may skip some if you are finding them easy) EXERCISE 14 C | QUESTIONS: P 261 -262 | ANSWERS: P 566 Q 3 -10 (you may skip some if you are finding them easy) 3

DEFINITE INTEGRALS Integration allows us to find the area underneath a curve. When we use limits, we call the integral a ‘definite integral’ In year 2, we look more closely at why integration finds the area under the curve. 25/05/2021 4

DEFINITE INTEGRALS | EXAMPLE-PROBLEM PAIRS 1 E. We substitute in the upper and lower limits Reducing Mistakes 25/05/2021 5

CHECKING YOUR ANSWER Reducing Mistakes Practice using your calculator the check your answer to every definite integral – make it a habit. 25/05/2021 6

DEFINITE INTEGRALS | EXAMPLE-PROBLEM PAIRS 1 P. 25/05/2021 7

DEFINITE INTEGRALS | EXAMPLE-PROBLEM PAIRS 2 E. 25/05/2021 8

DEFINITE INTEGRALS | EXERCISE Q 1 ✓/✗ 25/05/2021 9

DEFINITE INTEGRALS | EXERCISE Q 2 ✓/✗ 25/05/2021 10

DEFINITE INTEGRALS | EXERCISE Q 3 ✓/✗ 25/05/2021 11

DEFINITE INTEGRALS | EXERCISE Q 4 25/05/2021 12

DEFINITE INTEGRALS | EXERCISE Q 4 25/05/2021 13

EXTENSION QUESTION ✓/✗ 25/05/2021 ✓/✗ 14

25/05/2021 On mini-whiteboards: Find Can you make sense of this? 15

AREA BENEATH THE X-AXIS Area beneath the x-axis is ‘negative’ To calculate the shaded area, we need to do separate calculations for when the graph is above or below the x-axis. 25/05/2021 16

AREA BENEATH THE X-AXIS |EXAMPLE-PROBLEM PAIRS 25/05/2021 2 E. First we write down the roots: B A Example continues on next slide… 17

AREA BENEATH THE X-AXIS |EXAMPLE-PROBLEM PAIRS 25/05/2021 B A Use the absolute value for each separate area 18

AREA BENEATH THE X-AXIS |EXAMPLE-PROBLEM PAIRS 2 P. Roots: 25/05/2021 19

AREA BENEATH THE X-AXIS |EXAMPLE-PROBLEM PAIRS 25/05/2021 2 P. A B 20

AREA BENEATH THE X-AXIS |EXERCISE Q 1 25/05/2021 21

AREA BENEATH THE X-AXIS |EXERCISE Q 1 25/05/2021 22

AREA BENEATH THE X-AXIS |EXERCISE Q 2 25/05/2021 23

AREA BENEATH THE X-AXIS |EXERCISE Q 2 25/05/2021 24

AREA BENEATH THE X-AXIS |EXERCISE Q 3 25/05/2021 25

AREA BENEATH THE X-AXIS |EXERCISE Q 3 25/05/2021 26

AREA BETWEEN A LINE AND A CURVE To find the area between a line and a curve: 1 Find the points of intersection (these will be the limits for the integral) 25/05/2021 27

AREA BETWEEN A LINE AND A CURVE 2 Find the area underneath the line (in this case the area of a trapezium) 3 Find the area underneath the curve using integration (this is the part you don’t want). 4 Subtract the area under curve from the area under the line. 25/05/2021 28

AREA BETWEEN A LINE AND A CURVE | EXAMPLE-PROBLEM PAIRS 25/05/2021 1 E. Points of intersection: 13 8 R 1 Area under line (trapezium): 6 29

AREA BETWEEN A LINE AND A CURVE | EXAMPLE-PROBLEM PAIRS 25/05/2021 13 Area under curve: 8 R 1 Area of shaded region: 6 30

AREA BETWEEN A LINE AND A CURVE | EXAMPLE-PROBLEM PAIRS 25/05/2021 1 P. Points of intersection: 9 1 Area under line (trapezium): -3 1 31

AREA BETWEEN A LINE AND A CURVE | EXAMPLE-PROBLEM PAIRS 25/05/2021 1 P. Area under line (trapezium): 9 Area under curve: 1 -3 Area of shaded region: 1 32

AREA BETWEEN A LINE AND A CURVE | EXAMPLE-PROBLEM PAIRS 2 E. Points of intersection: 9 A B 3 Area of region A (triangle): 25/05/2021 33

AREA BETWEEN A LINE AND A CURVE | EXAMPLE-PROBLEM PAIRS 25/05/2021 2 E. Area of region B (under the curve): Upper limit: 9 B A 3 6 34

AREA BETWEEN A LINE AND A CURVE | EXERCISE 14 E | QUESTIONS: P 268 -272 | ANSWERS: P 567 Q 2 (a)(i) and (b)(ii) Q 5, Q 6, Q 10 -15 25/05/2021 35

AREA BETWEEN TWO CURVES 25/05/2021 Note: This topic does not appear in AS maths, but is in A 2 maths. As seen previously, if an integrand contains multiple terms, we can integrate each term separately. e. g. We also extend previous examples to find the shaded area in the diagram as follows: Combining these two facts, we can see that: We can subtract the expressions before integrating. 36

AREA BETWEEN TWO CURVES | EXAMPLE-PROBLEM PAIRS 1 E. Find the area of the shaded region. Roots: 25/05/2021 37

25/05/2021 2 E. Find the area of the shaded region. Roots: 38

AREA BETWEEN TWO CURVES | EXERCISE 12 F | QUESTIONS: P 251 -252 | ANSWERS: P 548 Q 1. (d)(i), (d)(ii) Q 3, Q 4, Q 5, Q 6 25/05/2021 39

EXPLORING AREAS FURTHER The area of the shaded region is 40 square units. Q 1 Which of the following can we evaluate? (a) (b) (c) (d) 25/05/2021 40

EXPLORING AREAS FURTHER The area of the shaded region is 40 square units. Q 1 Which of the following can we evaluate? (a) (b) (c) (d) 25/05/2021 41

EXPLANATION FOR PART (C) 25/05/2021 42

EXPLORING AREAS FURTHER The area of the shaded region is 40 square units. Q 1 Evaluate ✓/✗ 25/05/2021 43

EXPLANATION 25/05/2021 44

MIXED INTEGRATION PRACTICE | EXERCISE 25/05/2021 45

MIXED INTEGRATION PRACTICE | EXERCISE 25/05/2021 46

MIXED INTEGRATION PRACTICE | EXERCISE 25/05/2021 47