DP Maths AI HL 5.5 Integration and Its Applications

DP Maths AI HL 5.5 Integration and Its Applications - Full notes

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DP Maths AI HL | 5.5 Integration and Its Applications

DP MATHEMATICS AI · TOPIC 5: CALCULUS

5.5

Integration and Its Applications

Full notes for Applications and Interpretation HL

Contents

1	Accumulation and area
2	Anti-differentiation
3	Integrating powers of x
4	Sums, multiples and rewriting
5	Finding the constant of integration
6	Definite integrals
7	Definite integrals with technology
8	Area under a curve
9	Regions below the x-axis
10	Area between a curve and a line
11	The trapezoidal rule
12	The trapezoidal rule from data
13	Overestimates and underestimates
14	Application: distance from velocity
15	Application: totals from rates
16	Application: cost from marginal cost
17	Using technology for integration
18	Mixed review: worked problems
19	Further integrals and substitution (HL)
20	Volumes of revolution (HL)
21	Investigation: estimating the area of a lake
22	Exam-style questions with markscheme
23	Common misconceptions and links
24	Check your understanding
25	Summary and topic map
26	Answers to practice exercises

These are full teaching notes. Read each section in order, try the Think about it question before you read on, and work through every worked example before attempting the practice exercise.

Before you start

Rate yourself on each statement now, and again after you've worked through the notes. Shade R (not yet), A (getting there) or G (confident).

I can…	R	A	G
understand integration as anti-differentiation and accumulation
integrate axn (n ≠ −1) and sums of such terms
find the constant of integration from a known point
evaluate definite integrals with technology
find areas under curves, including regions below the axis
use the trapezoidal rule with functions and data
decide whether an estimate is an over- or underestimate
apply integration to distance, totals and costs

HL booklet: this booklet contains all the SL content plus integrals of further functions, substitution and volumes of revolution.

Guiding questions

How can a total be found from a rate of change? How can areas of irregular regions be estimated?

How this topic is examined

Integration appears on both papers: anti-differentiation and finding the constant (Paper 1), definite integrals and areas with technology (Paper 2), and the trapezoidal rule with data, including whether it over- or underestimates.

Write the integral you evaluate with its limits, state that technology was used, and give units for accumulated quantities (for example, m2, km, litres).

Mathematics: Applications and Interpretation places strong emphasis on technology, modelling and interpretation. Most papers allow a GDC, so you are expected to use it efficiently, record the method you used, and interpret every answer in context.

Command term	What the examiner wants
Write down	an answer with little or no working; it should be obtainable by inspection
Find / Calculate / Determine	obtain the answer showing relevant working, including what you entered into a GDC
Show that	obtain the given result; every step must be shown and the result must not be assumed
Estimate	obtain an approximate value
Sketch	a diagram showing the general shape and key features, labelled
Interpret / Comment	explain the meaning of a result in context, or give a judgement
Justify / Explain	give valid reasons or evidence
Hence	use the previous result

How to use these notes

Box	What it's for
Think about it	a question to get you thinking before a section
Why it works	a derivation or explanation of a key result
Key point	a result or idea you must not forget
Common mistake	an error examiners see every year, and how to avoid it
GDC tip	how to use your graphic display calculator efficiently
In the real world	where the mathematics is used outside the classroom
Worked example	a full solution with IB-style marks (M1 method, A1 answer, R1 reasoning, AG answer given), graded basic, standard and exam level
Practice exercise	graded questions for each section; full answers are at the end of the booklet

1 Accumulation and area

Adding up small pieces.

Think about it

A car travels at a constant 20 m/s for 10 s. How far does it go? How does this relate to the area under the velocity-time graph? What if the speed is changing?

If a rate is constant, total = rate × time, which is the area of a rectangle under the rate graph. If the rate varies, we split the time into thin strips, approximate each by a rectangle, and add. As the strips get thinner, the sum approaches the exact area under the curve: the definite integral. Integration therefore turns a rate of change into an accumulated total.

2 Anti-differentiation

Reversing the derivative.

If F′(x) = f(x), then F is an antiderivative of f. Since constants differentiate to zero, there is a family of antiderivatives F(x) + c. The indefinite integral is written ∫f(x) dx = F(x) + c.

3 Integrating powers of x

Add one to the power, divide by the new power.

WORKED EXAMPLE 1 Powers (basic)

Find (a) ∫x4 dx; (b) ∫6x2 dx; (c) ∫5 dx.

Solution

(a) x5/5 + cA1

(b) 2x3 + c; (c) 5x + cA1

Common mistake

Forgetting + c in indefinite integrals, or differentiating instead of integrating.

Practice exercise 1

1. Find ∫8x3 dx.

2. Find ∫(1/2)x dx.

3. Find ∫0.3x2 dx.

4 Sums, multiples and rewriting

Term by term.

Integrate each term separately. Rewrite roots and fractions as powers (√x = x1/2, 3/x2 = 3x−2) and expand brackets first.

WORKED EXAMPLE 2 Polynomial (standard)

Find ∫(3x2 − 4x + 7) dx.

Solution

= x3 − 2x2 + 7x + cA2

WORKED EXAMPLE 3 Rewriting (standard)

Find ∫(2/x3 + (x + 1)2) dx.

Solution

2x−3 + x2 + 2x + 1M1

= −x−2 + x3/3 + x2 + x + cA1

Practice exercise 2

1. Find ∫(x3 − 6x + 2) dx.

2. Find ∫4/x2 dx.

3. Find ∫x(x − 3) dx.

5 Finding the constant of integration

Use a known point.

WORKED EXAMPLE 4 Curve through a point (standard)

dy/dx = 6x2 − 2 and the curve passes through (1, 5). Find y.

Solution

y = 2x3 − 2x + cA1

5 = 2 − 2 + c: c = 5M1

y = 2x3 − 2x + 5A1

WORKED EXAMPLE 5 Cost from marginal cost (exam level)

C′(x) = 12 − 0.04x and the fixed cost is $500. Find C(x) and C(100).

Solution

C(x) = 12x − 0.02x2 + c, c = 500A1 A1

C(100) = 1200 − 200 + 500 = $1500A1

Practice exercise 3

1. f′(x) = 3x2 + 1, f(2) = 12. Find f(x).

2. The gradient of a curve is 4x − 5 and it passes through (2, 0). Find its y-intercept.

6 Definite integrals

Evaluate between limits.

WORKED EXAMPLE 6 Definite integral (standard)

Evaluate ∫ from 1 to 3 of (2x + 1) dx.

Solution

[x2 + x] from 1 to 3M1

= 12 − 2 = 10A1

Practice exercise 4

1. Evaluate ∫ from 0 to 2 of 3x2 dx.

2. Evaluate ∫ from 1 to 4 of (x2 − x) dx.

7 Definite integrals with technology

Numerical integration.

GDC tip

Use the numerical integration function (∫f(x)dx or fnInt). Enter the function, the variable, and the lower and upper limits. On a graph, the ∫f(x)dx tool shades and evaluates the area between limits.

WORKED EXAMPLE 7 GDC (standard)

Evaluate ∫ from 0 to 4 of 20e−0.3x dx.

Solution

GDCM1

= 46.6A1

Practice exercise 5

1. Evaluate ∫ from 0 to 3 of √(x2 + 1) dx.

2. Evaluate ∫ from 1 to 5 of 10/x dx.

8 Area under a curve

Between the curve and the x-axis.

If f(x) ≥ 0 on [a, b], the area between y = f(x), the x-axis and x = a, x = b is ∫ from a to b of f(x) dx.

Figure 1. Area under a curve between its roots.

WORKED EXAMPLE 8 Area (standard)

Find the area enclosed by y = −0.5x2 + 4x and the x-axis.

Solution

Roots x = 0 and 8A1

∫ from 0 to 8 = [−x3/6 + 2x2] = 42.7 (128/3)M1 A1

Practice exercise 6

1. Find the area under y = x2 + 1 from x = 0 to x = 3.

2. Find the area enclosed by y = 9 − x2 and the x-axis.

9 Regions below the x-axis

Areas are positive.

If the curve is below the x-axis, the integral is negative. To find a total area, split at the roots and add the absolute values, or integrate |f(x)| with technology.

WORKED EXAMPLE 9 Area crossing the axis (exam level)

Find the total area between y = x2 − 4 and the x-axis for 0 ≤ x ≤ 3.

Solution

∫ from 0 to 2 = −16/3; ∫ from 2 to 3 = 7/3M1 A1

Area = 16/3 + 7/3 = 23/3 ≈ 7.67A1

Practice exercise 7

1. Find the total area between y = x3 and the x-axis for −2 ≤ x ≤ 2.

10 Area between a curve and a line

Upper minus lower.

The area between two graphs is ∫(upper − lower) dx between their intersection points. On Paper 2, find the intersections with the GDC, then integrate the difference.

WORKED EXAMPLE 10 Curve and line (standard)

Find the area enclosed by y = x2 and y = 2x + 3.

Solution

Intersections x = −1, 3A1

∫ from −1 to 3 of (2x + 3 − x2) dxM1

= 32/3 ≈ 10.7A1

Practice exercise 8

1. Find the area between y = 4x − x2 and y = x.

11 The trapezoidal rule

Approximating with trapezia.

Divide [a, b] into n strips of equal width h; each strip is approximated by a trapezium. The rule is used when the function cannot be integrated, or when only data values are known.

Figure 2. Trapezia approximating the area.

WORKED EXAMPLE 11 Trapezoidal rule (standard)

Estimate ∫ from 0 to 4 of (1 + 0.5x2) dx with 4 strips.

Solution

h = 1; y = 1, 1.5, 3, 5.5, 9A1

½[1 + 9 + 2(1.5 + 3 + 5.5)] = 15M1 A1

Exact value 14.67: overestimateR1

Practice exercise 9

1. Estimate ∫ from 1 to 3 of 1/x dx with 4 strips.

2. Estimate ∫ from 0 to 2 of ex dx with h = 0.5.

12 The trapezoidal rule from data

When only measurements exist.

Distance across (m)	0	5	10	15	20	25	30
Depth (m)	0	1.4	2.6	3.1	2.8	1.5	0

WORKED EXAMPLE 12 River cross-section (exam level)

Estimate the cross-sectional area of the river.

Solution

h = 5A1

2.5[0 + 0 + 2(1.4 + 2.6 + 3.1 + 2.8 + 1.5)]M1

= 57 m2A1

Practice exercise 10

1. Water flows at 0.8 m/s through this cross-section. Estimate the volume per minute.

13 Overestimates and underestimates

Concavity decides.

If the curve is concave up (bending upwards), the trapezia lie above the curve, so the rule overestimates. If concave down, it underestimates. More strips give better estimates.

Practice exercise 11

1. Does the trapezoidal rule over- or underestimate ∫ from 0 to 2 of √x dx? Explain.

2. How can the estimate be improved?

14 Application: distance from velocity

Area under a velocity graph.

If v(t) is velocity, ∫ from t1 to t2 of v dt is the displacement; ∫|v| dt is the total distance travelled.

WORKED EXAMPLE 13 Distance (standard)

v(t) = 3t2 − 12t + 9 (m/s). Find the distance travelled for 0 ≤ t ≤ 4.

Solution

GDC: ∫ from 0 to 4 of |v| dtM1

= 12 m (displacement 4 m)A1 A1

Practice exercise 12

1. A runner's speed is v = 6 − 0.2t m/s for 0 ≤ t ≤ 20. Find the distance run.

15 Application: totals from rates

Flow, production, consumption.

WORKED EXAMPLE 14 Water flow (standard)

Water flows into a tank at r(t) = 40 + 6t − 0.3t2 litres per minute for 0 ≤ t ≤ 20. Find the total volume added.

Solution

∫ from 0 to 20 of r(t) dtM1

= 800 + 1200 − 800 = 1200 LA1

Practice exercise 13

1. Electricity use (kW) is P(t) = 2 + sin(πt/12) for 0 ≤ t ≤ 24 hours. Find the energy used in kWh.

16 Application: cost from marginal cost

Accumulating small costs.

WORKED EXAMPLE 15 Extra cost (standard)

C′(x) = 20 − 0.02x. Find the extra cost of increasing production from 200 to 300 items.

Solution

∫ from 200 to 300 of (20 − 0.02x) dxM1

= 2000 − 500 = $1500A1

Practice exercise 14

1. Marginal revenue R′(x) = 50 − 0.1x. Find the revenue from the first 100 items.

17 Using technology for integration

A checklist.

GDC tip

1. Graph the function to see where it is above or below the axis. 2. Find intersections or roots with the GDC. 3. Use numerical integration with the correct limits. 4. Write down the integral you evaluated in your working.

18 Mixed review: worked problems

Bringing the ideas together.

WORKED EXAMPLE 16 Mixed problem (exam level)

f′(x) = 3x2 − 6x, and f(0) = 4. Find f(x) and the area under y = f(x) for 0 ≤ x ≤ 2.

Solution

f(x) = x3 − 3x2 + 4A1 A1

∫ from 0 to 2 = [x4/4 − x3 + 4x] = 4 − 8 + 8 = 4M1 A1

Practice exercise 15

1. Find k if ∫ from 0 to k of 2x dx = 25 (k > 0).

19 Further integrals and substitution (HL)

Beyond powers.

f(x)	∫f(x) dx
1/x	ln\|x\| + c
ex	ex + c
sin x	−cos x + c
cos x	sin x + c
1/cos2x	tan x + c

For linear inside functions, divide by the coefficient: ∫e3x dx = e3x/3 + c. More generally, ∫f(g(x))g′(x) dx = F(g(x)) + c (substitution u = g(x)).

WORKED EXAMPLE 17 Substitution (exam level)

Find ∫2x(x2 + 1)4 dx.

Solution

u = x2 + 1, du = 2x dxM1

∫u4 du = u5/5A1

= (x2 + 1)5/5 + cA1

Practice exercise 16

1. Find ∫(4e2x + 3/x) dx.

2. Find ∫sin(3x) dx.

20 Volumes of revolution (HL)

Rotating a region.

WORKED EXAMPLE 18 Volume (standard)

The region under y = √x for 0 ≤ x ≤ 4 is rotated 2π about the x-axis. Find the volume.

Solution

π∫ from 0 to 4 of x dxM1

= 8π ≈ 25.1A1

Practice exercise 17

1. Find the volume when y = x2, 0 ≤ x ≤ 2, is rotated about the x-axis.

2. A vase is modelled by rotating y = 3 + sin(x/2), 0 ≤ x ≤ 20 cm, about the x-axis. Estimate its volume with technology.

21 Investigation: estimating the area of a lake

A Paper 3 style exploration.

A lake is surveyed by measuring its width (m) at 20 m intervals along a baseline: 0, 34, 52, 61, 58, 47, 30, 0. 1. Estimate the area using the trapezoidal rule. 2. Fit a polynomial model to the widths with technology and integrate it. 3. Compare the two estimates and discuss which is more reliable. 4. Explain how halving the interval would affect the estimate. 5. The lake's average depth is 3.2 m. Estimate its volume.

Answers. 1. 10[0 + 0 + 2(34 + 52 + 61 + 58 + 47 + 30)] = 5640 m2. 2. A quartic regression integrated over 0 ≤ x ≤ 140 gives about 5770 m2 (depends on the model). 3. The trapezoidal rule underestimates where the outline is concave down. 4. More strips give a better estimate. 5. About 18 000 m3.

22 Exam-style questions with markscheme

Attempt each question under timed conditions before reading the markscheme.

Question 1 (Paper 1 style) (about 10 minutes)

f′(x) = 4x3 − 6x + 1 and f(1) = 3.

(a) Find f(x). [4]

(b) Find ∫ from 0 to 1 of f′(x) dx. [2]

Markscheme

Part	Answer	Marks
(a)	x4 − 3x2 + x + c	A2
	3 = 1 − 3 + 1 + c	M1
	f(x) = x4 − 3x2 + x + 4	A1
(b)	f(1) − f(0)	M1
	−1	A1

Examiner's comments. Part (b) uses the fundamental theorem.

Question 2 (Paper 2 style) (about 12 minutes)

The speed of a cyclist (m/s) is recorded every 10 s: 0, 4.2, 6.8, 8.1, 8.6, 8.4, 7.9.

(a) Use the trapezoidal rule to estimate the distance in 60 s. [3]

(b) A model is v(t) = 9(1 − e−0.06t). Use it to estimate the distance. [2]

Markscheme

Part	Answer	Marks
(a)	5[0 + 7.9 + 2(4.2 + 6.8 + 8.1 + 8.6 + 8.4)]	M1 A1
	400.5 m	A1
(b)	∫ from 0 to 60 of v dt	M1
	394 m	A1
(c)	close agreement; the model does not show the slight decrease after 40 s	R1

Examiner's comments. Accept reasonable comments.

Question 3 (Paper 2 style) (about 8 minutes)

The curves y = x2 and y = 6 − x intersect at two points.

(a) Find the x-coordinates of the intersections. [2]

(b) Find the area enclosed. [2]

Markscheme

Part	Answer	Marks
(a)	x2 + x − 6 = 0	M1
	x = −3, 2	A1
(b)	∫ from −3 to 2 of (6 − x − x2) dx	M1
	125/6 ≈ 20.8	A1

Examiner's comments. Upper minus lower function.

23 Common misconceptions and links

Students often think…	Actually…
forget + c	include it
area = integral when below the axis	take the absolute value
trapezoidal rule always overestimates	depends on concavity
∫xn = xn+1	divide by n + 1

How 5.5 connects to the rest of the course

Topic	Connection
5.3 Derivatives	reverse process
5.1 Limits	sums of strips
5.14 Differential equations	solving by integration
3.1 Geometry	areas and volumes

24 Check your understanding

Section A: multiple choice

1. ∫6x2 dx =

A. 2x3 + c B. 12x + c C. 6x3 + c D. 3x2 + c

2. ∫ from 0 to 3 of 2x dx =

A. 6 B. 3 C. 9 D. 18

3. The trapezoidal rule with a concave up curve

A. underestimates B. overestimates C. is exact D. fails

4. Displacement from velocity is

A. dv/dt B. v × t always C. ∫a dv D. ∫v dt

5. h for 6 strips on [0, 3] is

A. 6 B. 0.5 C. 3 D. 2

Section B: structured

6. Find the area under y = 3x2 + 2 from x = 1 to x = 2. [2]

7. Use 3 strips to estimate ∫ from 0 to 3 of x3 dx. [3]

Answers

1. A 2. C 3. B 4. D 5. B

6. 9

7. 0.5[0 + 27 + 2(1 + 8)] = 22.5

25 Summary and topic map

Idea	Remember
Antiderivative	∫axn dx = axn+1/(n + 1) + c
Constant	from a known point
Definite	F(b) − F(a); GDC
Area	split at roots; upper − lower
Trapezoidal rule	(h/2)[y0 + yn + 2(middle)]
Estimates	concave up over, concave down under
Applications	distance, totals, costs

Before the exam, make sure you…

•include + c
•split areas at roots
•state the integral used with the GDC
•give units

Integration and Its Applications on one page

Cover this page with a sheet of paper and try to rebuild it from memory. Whatever you can't recall is what to revise next.

Figure 3. Topic map for Integration and Its Applications.

26 Answers to practice exercises

Check your working against these answers. Full methods are expected in the exam.

Practice exercise 1

1. 2x4 + c

2. x2/4 + c

3. 0.1x3 + c

Practice exercise 2

1. x4/4 − 3x2 + 2x + c

2. −4/x + c

3. x3/3 − 3x2/2 + c

Practice exercise 3

1. x3 + x + 2

2. y = 2x2 − 5x + 2: (0, 2)

Practice exercise 4

1. 8

2. 13.5

Practice exercise 5

1. 5.65

2. 16.1

Practice exercise 6

1. 12

2. 36

Practice exercise 7

1. 8

Practice exercise 8

1. 4.5

Practice exercise 9

1. h = 0.5: 1.117

2. 6.52

Practice exercise 10

1. 57 × 0.8 × 60 = 2736 m3

Practice exercise 11

1. underestimate: √x is concave down

2. use more (narrower) strips

Practice exercise 12

1. 120 − 40 = 80 m

Practice exercise 13

1. 48 kWh

Practice exercise 14

1. $4500

Practice exercise 15

1. k = 5

Practice exercise 16

1. 2e2x + 3 ln|x| + c

2. −cos(3x)/3 + c

Practice exercise 17

1. 32π/5 = 20.1

2. ≈ 665 cm3

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