IB Mathematics: Applications and Interpretation SL — Topic 5 Calculus

Integration and Area

Connect anti-differentiation, boundary conditions, definite integrals and positive area using technology appropriately.

AI SL · SL 5.5

Learning goal

Anti-differentiate polynomial functions, determine constants from boundary conditions and evaluate definite integrals/areas using technology.

AI SL: SL 5.5 · Topic 5 Calculus.

Integration reverses differentiation and accumulates change; a boundary condition selects one antiderivative from a family.

\(\int ax^n\,dx=\frac{a}{n+1}x^{n+1}+C\quad(n\ne-1)\)

Integration reverses differentiation and accumulates change; a boundary condition selects one antiderivative from a family.

Anti-differentiate polynomial functions, determine constants from boundary conditions and evaluate definite integrals/areas using technology.

\[\int ax^n\,dx=\frac{a}{n+1}x^{n+1}+C\quad(n\ne-1)\]

\[\int_a^b f(x)\,dx=F(b)-F(a)\]

\[A=\int_a^b f(x)\,dx\quad\text{when }f(x)>0\]

For an indefinite integral, include \(+C\).
Use a boundary condition to determine \(C\) when required.
For an area question, first write the integral expression, then evaluate it with the GDC if appropriate.

Question

Given \(f\'(x)=3x^2+x\) and \(f(1)=10\), find \(f(x)\).

Solution

Answer: \(f(x)=x^3+\tfrac12x^2+\tfrac{17}{2}\).

Try these questions before moving on.

Find an antiderivative of \(5x^4-2x\).
Use a boundary condition to determine \(C\).
Write the definite integral that represents the area under a positive curve from \(x=a\) to \(x=b\).

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