Functions and algebra: Analyse and represent mathematical and contextual situations using integrals and find areas under curves by using integration rules
Subject outcome
Subject outcome 2.5: Analyse and represent mathematical and contextual situations using integrals and find areas under curves by using integration rules
Learning outcomes
- Find the integrals of the following:
[latex]\scriptsize \int a{{x}^{n}}dx[/latex] ; [latex]\scriptsize \int \displaystyle \frac{a}{x}dx[/latex] ; [latex]\scriptsize \int a{{e}^{{kx}}}dx[/latex] ; [latex]\scriptsize \int a\text{sin}kxdx[/latex] ; [latex]\scriptsize \int a\text{cos}kxdx[/latex] ; [latex]\scriptsize \int \text{ase}{{\text{c}}^{2}}kxdx[/latex]
Where:
[latex]\scriptsize \int a{{x}^{n}}dx=\displaystyle \frac{{a{{x}^{{n+1}}}}}{{n+1}}+c[/latex]
[latex]\scriptsize \int \displaystyle \frac{a}{x}dx=a\ln x+c[/latex]
[latex]\scriptsize \int a{{e}^{{kx}}}dx=\displaystyle \frac{{a{{e}^{{kx}}}}}{k}+c[/latex]
[latex]\scriptsize \int a\text{sin}kxdx=\displaystyle \frac{{-a\text{cos}kx}}{k}+c[/latex]
[latex]\scriptsize \int a\text{cos}kxdx=\displaystyle \frac{{a\text{sin}kx}}{k}+c[/latex]
Note:- Simplifications may be required where necessary.
- Integrals of polynomials may be assessed.
- Integration by parts is excluded.
- Use the upper and lower limits to calculate definite integrals.
- Determine the area under a curve by:
- Working from a given graph or by sketching a graph.
- Working with an area bounded by a curve, the x-axis, an upper and a lower limit.
- Splitting the area into two intervals when the graph crosses the x-axis.
Note:- Integrals with respect to the x-axis only.
- Areas between two curves are excluded.
- The y-axis (x = 0) may be used as an upper or lower limit.
Unit 1 outcomes
By the end of this unit you will be able to:
- Understand the relationship between differentiation and integration (anti-derivative).
- Define integration as the approximate area under curves.
Unit 2 outcomes
By the end of this unit you will be able to:
- Apply the rules of integration to various functions.
- Calculate the definite integral.
Unit 3 outcomes
By the end of this unit you will be able to:
- Find the area under a curve between two points and bound by the x-axis.
