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Integral calculus, grade 11 (5 units)

When you know how fast a quantity changes – the speed of a car, the rate at which water flows into a pool – you can find how much of it has accumulated.Subjects

That is integral calculus: the reverse of the differential calculus you learnt in grade 10 and in grade 11’s differential calculus course (the derivatives of the trigonometric functions, the second derivative and concavity). The topics, in order:

  • the antiderivative: the indefinite integral of polynomials, of powers and of roots;
  • accumulation: a constant rate and a step function, approximating with rectangles, the accumulation function and its properties, the fundamental theorem of calculus, the definite integral;
  • the graphs of a function, of its derivative and of its accumulation function;
  • the rules of integration: a constant factor, a sum, a linear inner function, a composite function times its inner derivative, the trigonometric functions and integrals by identities;
  • applications: a function from its derivative and a point, the area between a graph and the x\displaystyle x-axis and between two graphs, accumulation problems – distance, water, a parameter, an extremum.

There are two kinds of lessons:

  • learning lessons – an explanation, worked examples and a few exercises to try;
  • practice lessons – questions only.

How the page works: Using a page.

  1. The antiderivative and the indefinite integralLearningAn antiderivative and the family of antiderivatives $F(x)+C$; the sign $\int f(x)\,dx$; the integral of a power and of a polynomial; checking by differentiating; the antiderivative through a point.
  2. Integrals of negative powers and of rootsLearning$\frac{1}{x^n}$ as a negative power and $\sqrt{x}$ as the power $\frac{1}{2}$; the integrals of $\frac{1}{x^2}$, $\sqrt{x}$ and $\frac{1}{\sqrt{x}}$; splitting a fraction into a sum before integrating.
  3. Accumulation at a constant rate and of a step functionLearningA rate of change and the accumulated quantity: distance from speed, water from a rate of flow; the accumulation is the area under the graph of the rate; a negative rate – the area counted as negative; a step function; displacement and distance.
  4. Approximating the accumulation with rectanglesLearningA rate that is not constant: split the interval into pieces, and in each piece replace the function with a constant value – a rectangle; the value at the left end, the right end and their mean; the shorter the pieces, the better the approximation; a linear function – the mean gives the exact area.
  5. The accumulation function and its propertiesLearningThe accumulation function $F_a(x)=\int_a^x f(t)\,dt$ – the signed area from $a$ to $x$; $F_a(a)=0$; rising and falling by the sign of $f$, extremum points at the zeros where $f$ changes sign, concavity by the rise and fall of $f$, an inflection point at an extremum of $f$; another lower limit shifts the graph.
  6. The fundamental theorem of calculusLearningThe derivative of the accumulation function is the given function: $F_a'(x)=f(x)$ – the proof; the area under a line as an antiderivative; the accumulation function is the antiderivative that is 0 at $a$; $\int_a^b f(x)\,dx=F(b)-F(a)$.
  7. The definite integral and its propertiesLearningCalculating a definite integral with $\left[F(x)\right]_a^b$; a positive, a negative and a zero integral – by the signed areas; a constant factor, a sum, joining intervals, swapping the limits; shifting the function and shifting the interval; an integral when only an area is given.
  8. Practice: accumulation and the definite integralPracticeIndefinite integrals of powers and roots; the accumulation of a step function; approximating with rectangles; the accumulation function from a graph and from a formula; the fundamental theorem; definite integrals and their properties; a parameter in a limit.
  9. The graphs of a function, its derivative and its accumulation functionLearning$F'=f$ and $F''=f'$: from the sign of $f$ – the rise, fall and extrema of $F$; from the sign of $f'$ – the concavity of $F$ and its inflection points; from a graph of $F$ to $f$ and from a graph of $f'$ to $f$; the accumulation function from another lower limit.
  10. The integral of a composite function with a linear inner functionLearning$\int f(mx+b)\,dx=\frac{1}{m}F(mx+b)+C$ – from the chain rule; powers, roots and fractions of $mx+b$; a definite integral; when the inner function is not linear – expand.
  11. The integral of a composite function times its inner derivativeLearning$\int f(u(x))\cdot u'(x)\,dx=F(u(x))+C$: the antiderivative is a composite function; spotting the inner derivative; making up a missing factor; a power and a root of a function times its derivative; a definite integral.
  12. Integrals of trigonometric functionsLearningThe integrals of $\cos x$, $\sin x$ and $\frac{1}{\cos^2x}$, in radians; a linear argument, $\sin(mx+b)$; an inner derivative, $\sin^nx\cos x$; a definite integral and the area of one arch; a whole period – an integral of 0.
  13. Integrals by trigonometric identitiesLearning$\sin^2x$ and $\cos^2x$ by the double angle; $\sin x\cos x=\frac{1}{2}\sin2x$; $\tan^2x=\frac{1}{\cos^2x}-1$; $(\sin x+\cos x)^2$; definite integrals and symmetry.
  14. Practice: the rules of integrationPracticeA linear inner function; a composite function times its inner derivative; integrals of trigonometric functions and by identities; definite integrals; an accumulation function and an equation.
  15. A function from its derivative and a pointLearningOf all the antiderivatives, a point on the graph picks one; a point from a tangent or from an extremum point; a function from its second derivative – two constants and two conditions; position from velocity; a function from the graph of its derivative.
  16. The area between a graph and the x-axisLearningAn area above the axis and below it; a function that changes sign – split at the zeros; the area enclosed by a graph and the axis; the area of a trigonometric function; an integral compared with an area.
  17. The area between two graphsLearning$\int_a^b\left(f(x)-g(x)\right)dx$ where $f\geq g$ – below the axis too; the meeting points as limits; graphs that swap places; a composite area – the upper edge passes from one graph to another; trigonometric graphs.
  18. Accumulation problems: motion, flow, a parameter and an extremumLearningDisplacement and distance from a velocity function; an amount accumulated from a rate of flow; a rate from a graph; a parameter fixed by a given area; an area as a function of a limit, and its extremum by the fundamental theorem.
  19. Practice: a function from its derivative, areas and accumulationPracticeA function from its derivative and a point and from its second derivative; the area between a graph and the axis; the area between two graphs, trigonometric ones too; displacement and distance; a parameter from an area.
  20. Review: integral calculusPracticeA summary of the course, and questions that combine all its topics: a trigonometric function and its derivative, the area between a graph and a line; an accumulation function from a graph, extrema and inflection; a function from its derivative and a tangent; the area between a graph, a tangent and an axis.
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