Advanced Calculus - Integration

This lesson comprises three (3) master classes focusing on:

• Anti-derivative
• Indefinite and definite integrals
• Reverse-chain rule
• Area under a curve
• Trapezoidal rule
• Fundamental Theorem of Calculus

Content:

MA-C4.1

• Define anti-differentiation as the reverse of differentiation and use the notation \( \int f(x)dx \) for anti-derivatives or indefinite integrals
• Recognise that any two anti-derivatives of \( f(x) \) differ by a constant
• Establish and use the formula \( \int x^ndx=\frac{1}{n+1}x^{n+1}+c \), for \( n \ne −1 \)
• Establish and use the formula \( \int f^{\prime}(x)[f(x)]^ndx=\frac{1}{n+1}[f(x)]^{n+1}+c  \) where \( n \ne −1 \) (the reverse chain rule)
• Establish and use the formulae for the anti-derivatives of \( \sin (ax+b) \), \( \cos (ax+b) \) and \( \sec^2 (ax+b) \)
• Establish and use the formulae \( \int e^xdx=e^x+c \) and \( \int e^{ax+b}dx=\frac{1}{a}e^{ax+b}+c \)
• Establish and use the formulae \( \int \frac{1}{x}dx=\ln |x|+c \) and \( \int \frac{f^{\prime}(x)}{f(x)}dx=\ln |f(x)|+c \) for \( x \ne 0 \), \( f(x) \ne 0 \), respectively
• Establish and use the formulae \( \int a^xdx=\frac{a^x}{\ln a}+c \)
• Recognise and use linearity of anti-differentiation
  • examine families of anti-derivatives of a given function graphically
• Determine indefinite integrals of the form \( \int f(ax+b)dx \)
• Determine \( f(x) \), given \( f^{\prime}(x) \) and an initial condition \( f(a)=b \) in a range of practical and abstract applications including coordinate geometry, business and science

MA-C4.2

• Know that ‘the area under a curve’ refers to the area between a function and the x-axis, bounded by two values of the independent variable and interpret the area under a curve in a variety of contexts
• Determine the approximate area under a curve using a variety of shapes including squares, rectangles (inner and outer rectangles), triangles or trapezia
  • consider functions which cannot be integrated in the scope of this syllabus, for example \( f(x)=\ln x \), and explore the effect of increasing the number of shapes used
• Use the notation of the definite integral \( \int_a^b f(x)dx \) for the area under the curve \( y=f(x) \) from \( x=a \) to \( x=b \) if \( f(x) \ge 0 \)
• Use the Trapezoidal rule to estimate areas under curves
  • use geometric arguments (rather than substitution into a given formula) to approximate a definite integral of the form \( \int_a^b f(x)dx \), where \( f(x) \ge 0 \), on the interval \( a \le x \le b \), by dividing the area into a given number of trapezia with equal widths
  • demonstrate understanding of the formula: \( \int_a^b f(x)dx \approx \frac{b−a}{2n}[f(a)+f(b)+2\{f(x_1)+...+f(x_{n−1}\}] \) where \( a=x_0 \) and \( b=x_n \), and the values of \( x_0,x_1,x_2, \dots ,x_n \) are found by dividing the interval \( a \le x \le b \) into \( n \) equal sub-intervals
• Use geometric ideas to find the definite integral \( \int_a^b f(x)dx \)  where \( f(x) \) is positive throughout an interval \( a \le x \le b \)  and the shape of \( f(x) \) allows such calculations, for example when \( f(x) \) is a straight line in the interval or \( f(x) \) is a semicircle in the interval
• Understand the relationship of position to signed areas, namely that the signed area above the horizontal axis is positive and the signed area below the horizontal axis is negative
• Using technology or otherwise, investigate the link between the anti-derivative and the area under a curve
  • interpret \( \int_a^b f(x)dx \) as a sum of signed areas
  • understand the concept of the signed area function \( F(x)=\int_a^x f(t)dt \)
• use the formula \( \int_a^b f(x)dx=F(b)−F(a) \), where \( F(x) \) is the anti-derivative of \( f(x) \), to calculate definite integrals
  • understand and use the Fundamental Theorem of Calculus, \( F^{\prime}(x)=\frac{d}{dx}[\int_a^x f(t)dt]=f(x) \) and illustrate its proof geometrically
  • use symmetry properties of even and odd functions to simplify calculations of area
  • recognise and use the additivity and linearity of definite integrals
  • calculate total change by integrating instantaneous rate of change
• Calculate the area under a curve
• Calculate areas between curves determined by any functions within the scope of this syllabus
• Integrate functions and find indefinite or definite integrals and apply this technique to solving practical problems