Integral Calculus Chapter 8 Comparing functions Similarities of derivatives:

Differences in functions: Think: How does the vertical translation affect the antiderivative? How can the vertical translation be represented in indefinite

integral notation? Notation: Important notation: Not commonly used but needs to be understood Basic antiderivatives The following antiderivatives are covered in Maths Methods 3 and 4

where where For For Additionally for The definite integral Fundamental Theorem of Calculus Continuous functionon an

interval Denotes the signed area enclosed by the graph of -axis and lines and Where is any antiderivative of Please note: in the expression , the number is the lower limit of integration and is the upper limit of integration. Function is called the integrand

Notes: If for all , the area between and is given by If for all , the area between and is given by Properties of the definite integral Questions Complete these in your notes Find an antiderivative of

each of the following: Evaluate each of the following integrals: For the questions below: a) Find an antiderivative of b) Evaluate c) Evaluate

Working out space The fundamental theorem of calculus Review from Maths Methods 3 and 4 Signed Area Regions above the x-axis have positive signed area Regions below the x-axis have negative signed area Total area of the shaded region is

Total signed area of shaded region is The fundamental theorem of calculus Signed Area continued Consider the graph of shown to the right. Find the area of and Total Area: Signed Area: Examples Example

The graph of is shown below Find the area of the shaded region. Example Find the area under the graph of between and . Example Example Sketch the graph of . Shade the region for the area determined by and find this area.

Example The graph of is shown. Find the area of the shaded region. Questions Complete the following questions in your book Question 1 Question 3 The graph of is as shown. Find the area of the shaded region.

Consider the graph of . a) Find the coordinates of the intercepts with the axes b) Find the equations of all asymptotes c) Sketch the graph d) Find the area bound by the curve, the axis and the line Question 4 Question 2 Sketch the graph of . Shade the region for which the area is determined by the integral and evaluate this integral.

Find the area between the curve ., the xaxis and the linesand Questions Complete the following questions in your book Question 5 Question 6 The graph of for is as shown. Find the area of thee shaded region. a) Show that the curve has only one turning point.

b) Find the coordinates of this point and determine its nature. c) Sketch the curve. d) Find the area of the region enclosed by the curve and line . Area under the curve 8.2 Connections to the Study Design: AOS 3 Calculus Applications of Integration

Application of integration, arc lengths of curves, areas of regions bounded by curves and volumes of solids of revolution and a region about either coordinate axis Key Vocabulary: Argand Diagrams Complex Number System

Cartesian Coordinate system Imaginary unit Quadratic Equations Complex Conjugate Roots Distinct Rectangular form Cartesian form

Constant Denominator Surds Equating Powers Rotation Anticlockwise Key Notation Integration Recap Maths Methods

Trigonometric Functions Area bound by a curve and the -axis Recall the definite integral Gives measure of area bound by curve , -axis and lines and . Where , and k and c are constants

Exponential Functions Where , and k and c are constants Area bounded by a curve and the -axis Fundamental Theorem of Calculus Recall that the definite integral Measures the area bound by the curve the x-axis and the lines and Example Worked Example Find the area bound by the line

The x-axis and the lines and Example Using symmetry Worked Example Find the area bound by the curve , the -axis and the lines Example Areas involving basic trigonometric functions Worked Example Find the area under one arch of the sine curve

Example Areas involving basic exponential functions Worked Example Find the area bound by the coordinate axes, the graph of the and the line Area involving signed areas Areas above the -axis Areas below the -axis

for where for where Definite integral represents is positive: | =

| Definite integral represents is positive: ( ) = ( ) = ( )

Example Worked Example Find the area bound by the curve and the -axis Areas both above and below the -axis Evaluate each area separately and Required area:

Example Worked Example Find the area bound by the curve and the -axis Area between curves Let f and g be continuous functions on the interval such that For all Area of region bound by two curves and line and can be found by evaluating.

Example Worked Example Find the area between the parabola and the straight line Antiderivatives involving inverse circular function Connections to the Study Design: AOS 3 Calculus Techniques of Anti-differentiation

Techniques of anti-differentiation and for the evaluation of definite integrals: Anti-differentiation of to obtain Anti-differentiation of and by recognition that they are derivatives of corresponding inverse circular functions Use of the substitution to anti-differentiate expressions

Use of the trigonometric identities , in anti-differentiation techniques Anti-differentiation using partial fractions of rational functions Key Vocabulary: Key Notation

Differentiation Recap of Circular Functions Integration of Inverse Circular Functions Examples Find an antiderivative of each of the following: Evaluate each of the following

definite integrals: Other Substitution 8.4 Connections to the Study Design: AOS 3 Calculus Techniques of Anti-differentiation Techniques of anti-differentiation and for the evaluation of definite integrals: Anti-differentiation of to obtain

Anti-differentiation of and by recognition that they are derivatives of corresponding inverse circular functions Use of the substitution to anti-differentiate expressions Use of the trigonometric identities , in anti-differentiation techniques

Anti-differentiation using partial fractions of rational functions Key Vocabulary: Key Notation Non-linear substitution Tip for non-linear substitution: Reduce integrand to one of the

standard u forms shown in the table to the right. Remember: after making a substitution or original variable should be eliminated. Integral must be entirely in terms of new variable .

Integration - Non-linear substitution Trigonometric Functions Logarithmic functions Integrals involving the logarithmic function

When we have the special case Result of trigonometric functions: Exponential Functions Result of exponential functions:

Examples a) Find b) Find c) Find d) Find e) Find Linear Substitutions 8.3 Connections to the Study Design: AOS 3 Calculus Techniques of Anti-differentiation

Techniques of anti-differentiation and for the evaluation of definite integrals: Anti-differentiation of to obtain Anti-differentiation of and by recognition that they are derivatives of corresponding inverse circular functions Use of the substitution to anti-differentiate expressions

Use of the trigonometric identities , in anti-differentiation techniques Anti-differentiation using partial fractions of rational functions Key Vocabulary: Argand Diagrams

Complex Number System Cartesian Coordinate system Imaginary unit Quadratic Equations Complex Conjugate Roots Distinct

Rectangular form Cartesian form Constant Denominator Surds Equating Powers Rotation Anticlockwise Key Notation

Integration by substitution Example Finding integrals in the form of where Find Finding particular integrals of the form where The gradient of a curve is given by . Find the particular curve that passes through the origin.

Integration by substitution Finding integrals in the form of where Where , and Since Antidifferentiate Finding integrals in the form of where Evaluating definite integrals using a

linear substitution Example Evaluate Note: When we evaluate a definite integral the result is a number. When using substitution, change the terminals/limits to the new variable and evaluate this definite integral in terms of the new variable with new terminals.

Integrals of powers of trigonometric functions 8.5 Connections to the Study Design: AOS 3 Calculus Techniques of Anti-differentiation Techniques of anti-differentiation and for the evaluation of definite integrals: Anti-differentiation of to obtain

Anti-differentiation of and by recognition that they are derivatives of corresponding inverse circular functions Use of the substitution to anti-differentiate expressions Use of the trigonometric identities , in anti-differentiation techniques

Anti-differentiation using partial fractions of rational functions Key Vocabulary: Key Notation Products of sines and cosines Integrals of the form , where and are non-negative integers, can be considered in the following three cases. Case A: The power of sine is odd If is odd, write . Then:

and the substitution can now be made. Case B: The power of cosine is odd If is even and is odd, write . Then: and the substitution can now be made. Case C: Both powers are even If both m and n are even, then the identity or Also note that:

The identity is used in the following example. Example Find: Partial Fractions Connections to the Study Design: AOS 3 Calculus Techniques of Anti-differentiation Techniques of anti-differentiation and for the evaluation of definite integrals:

Anti-differentiation of to obtain Anti-differentiation of and by recognition that they are derivatives of corresponding inverse circular functions Use of the substitution to anti-differentiate expressions

Use of the trigonometric identities , in anti-differentiation techniques Anti-differentiation using partial fractions of rational functions Key Vocabulary: Key Notation Rational Function - Recap

If and are polynomials, then the below is a rational function: If degree of is less than the degree of , then is a proper fraction If degree of is greater than or

equal to the degree of , then is a improper fraction A rational function may be expressed as a sum of simpler functions by resolving the function into partial fractions. Example: Proper Fractions

Linear factor in the denominator, there will be a partial fraction of the form . For every repeated linear factor in the denominator, there will be partial fraction of the form and . For every irreducible quadratic factor in the denominator, there will be a

partial fraction of the form . Note: A quadratic expression is said to be irreducible if it cannot be factorised over . Steps to resolve algebraic fraction into its partial fractions Step 1: Write a statement of identity between the original fraction and a sum of appropriate number of partial fractions

Step 2: Express the sum of the partial fractions as a single fraction, and note that the numerators of both sides are equivalent Step 3: Find the vales of the introduced constants . By substituting appropriate values for or by equating coefficients Improper fractions

Improper algebraic fractions can be expressed as a sum of partial fractions by first diving the denominator into the numerator to produce a quotient and a proper fraction. This proper fraction can then be resolved into its partial fractions using the techniques just introduced. Examples Improper Fractions

Proper Fractions Resolve into partial fractions Resolve into partial fractions Resolve into partial fractions

Express as partial fractions Summary Examples of resolving a proper fraction into partial fractions Distinct linear factors

Repeated linear factors Irreducible quadratic factor Using partial fractions for integration Distinct Linear Factors Find

Improper Fractions Find Repeated Linear Factor Express in partial fractions and hence find Irreducible Quadratic Factor Find an antiderivative of by first expressing it as partial fractions.