IB Mathematics: Applications and Interpretation SL — Topic 5 Calculus

Anti-differentiation, Definite Integrals and Area

Reverse differentiation, use boundary conditions, evaluate definite integrals with technology and interpret area and accumulation.

AI SL · SL 5.5

Learning goal

Understand integration as the reverse of differentiation, use a boundary condition to find a particular function, use technology to evaluate definite integrals, and calculate the area under a curve when the function is positive.

Syllabus link

IB Mathematics AI SL: SL 5.5. Anti-differentiation; boundary conditions; definite integrals using technology; area enclosed by a positive curve and the x-axis.

Big idea

A derivative describes a rate of change. Integration rebuilds the original function or accumulates a total amount.

Key relationship

If \(F'(x)=f(x)\), then \(\int f(x)\,dx=F(x)+C\). For a positive curve, \(A=\int_a^b f(x)\,dx\).

Integration as anti-differentiation

If differentiation asks “What is the gradient function?”, integration asks “What function could have produced this derivative?”

Key notation

If \(F'(x)=f(x)\), then an anti-derivative of \(f\) is \(F\), and

\[\int f(x)\,dx=F(x)+C.\]

The \(+C\) is essential because infinitely many functions can have the same derivative.

\[\boxed{\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\qquad n\ne-1}\]

Power-rule habit

For each power of \(x\): increase the exponent by 1, then divide by the new exponent. This works for positive and negative integer powers, provided the original exponent is not \(-1\).

Worked example 1A: positive integer power

Find \(\int 5x^3\,dx\).

Solution

\[\begin{aligned}\int5x^3\,dx&=5\int x^3\,dx\\&=5\left(\frac{x^{3+1}}{3+1}\right)+C\\&=\frac54x^4+C.\end{aligned}\]

Check: \(\frac{d}{dx}(\frac54x^4)=5x^3\).

Worked example 1B: negative integer power

Find \(\int 4x^{-3}\,dx\).

Solution

\[\begin{aligned}\int4x^{-3}\,dx&=4\left(\frac{x^{-2}}{-2}\right)+C\\&=-2x^{-2}+C\\&=-\frac{2}{x^2}+C.\end{aligned}\]

Check: \(\frac{d}{dx}(-2x^{-2})=4x^{-3}\).

Worked example 1C: combining several terms

Find \(\int(6x^2-4x+5)\,dx\).

Solution

Make the hidden powers explicit: \(-4x=-4x^1\) and \(5=5x^0\). Integrate term by term:

\[\int(6x^2-4x+5)\,dx=2x^3-2x^2+5x+C.\]

Differentiating the result gives \(6x^2-4x+5\), confirming the anti-derivative.

Common trap

Every indefinite integral needs \(+C\). For example, \(\int(6x-4)\,dx=3x^2-4x+C\), not just \(3x^2-4x\).

Using a boundary condition

A derivative determines a family of functions. A point or other boundary condition allows us to determine the constant \(C\).

Boundary-condition routine

Anti-differentiate to obtain a general function containing \(C\).
Substitute the given point or value.
Solve for \(C\), then write the particular function.

Worked example 2: using a boundary condition

Given \(\frac{dy}{dx}=3x^2+x\) and \(y=10\) when \(x=1\), find \(y\) in terms of \(x\).

Solution

First anti-differentiate:

\[y=\int(3x^2+x)\,dx=x^3+\frac12x^2+C.\]

Now use the boundary condition:

\[10=1^3+\frac12(1)^2+C=1.5+C,\]

so \(C=8.5\). Therefore

\[\boxed{y=x^3+\frac12x^2+8.5}\]

Definite integrals using technology

A definite integral accumulates a total amount over an interval. In AI SL 5.5, you should be able to write the correct definite integral and then use technology to evaluate it.

Meaning of the notation

\[\int_a^b f(x)\,dx\]

means that values of \(f(x)\) are accumulated from \(x=a\) to \(x=b\). In context this may represent area, total change, distance, volume, money or another accumulated quantity.

Calculator habit

Before pressing buttons, identify the integrand, variable, lower limit, upper limit, and the meaning of the answer.

TI-Nspire CX tutorial: definite integrals

The finalized handout links this tutorial for the first calculator workflow. It shows how to enter a definite integral using the TI-Nspire CX Scratchpad.

Open the tutorial on YouTube →

TI-Nspire CX walkthrough: entering a definite integral

The handout demonstrates \(\int_1^5(3x^2+4)\,dx\) in six calculator steps.

1. Start with a clear Scratchpad.

2. Open Menu and choose Calculus.

3. Choose Numerical Integral.

4. The integral template appears.

5. Enter limits, integrand and variable.

6. Press Enter: the result is \(140\).

Worked example 3: evaluating a definite integral on the GDC

Use technology to evaluate \(\int_1^4(x^2+2x)\,dx\).

Solution

First write the correct expression, then enter the integrand \(x^2+2x\), the variable \(x\), and the limits 1 and 4.

\[\int_1^4(x^2+2x)\,dx=36.\]

Answer The accumulated total over \([1,4]\) is 36 units.

TI-Nspire CX evaluating the integral from 1 to 4 of x squared plus 2x — The mathematical integral should still be written before the calculator result.

Area under a positive curve

When the curve lies above the x-axis on the whole interval, the definite integral directly gives the geometrical area.

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

Important AI SL condition

At AI SL 5.5, this area formula is used when the curve stays above the x-axis. Questions involving sign changes and total geometrical area require additional ideas developed elsewhere.

TI-Nspire CX tutorial: area under a curve

The handout places this tutorial before the first area example. It demonstrates the graphical calculator method for finding the area between a positive curve and the x-axis.

Open the tutorial on YouTube →

Worked example 4: area enclosed by a parabola and the x-axis

Find the area enclosed by \(y=2x-x^2\) and the x-axis.

Solution

First find the intercepts:

\[2x-x^2=0\Rightarrow x(2-x)=0\Rightarrow x=0,2.\]

On \(0\le x\le2\), the curve is above the axis, so

\[A=\int_0^2(2x-x^2)\,dx\approx1.333.\]

TI-Nspire graph with shaded area under y equals 2x minus x squared — Graph view: the shaded region is the required area.

TI-Nspire integral from 0 to 2 of 2x minus x squared — Numerical integral view gives approximately 1.33333.

Answer Approximately \(1.333\) square units.

Worked example 5: area over a stated interval

Find the area between \(y=3x+4\) and the x-axis from \(x=2\) to \(x=4\).

Solution

Since \(3x+4>0\) on \([2,4]\),

\[A=\int_2^4(3x+4)\,dx=26.\]

TI-Nspire graph showing the area under y equals 3x plus 4 from x equals 2 to 4 — Graph/enclosed-area method.

TI-Nspire numerical integral of 3x plus 4 from 2 to 4 — Numerical integral method.

Answer \(26\) square units.

Interpreting a definite integral in context

The shaded region on a graph is geometrical, but the units and meaning of a definite integral depend on the quantities shown on the axes.

Units of an accumulated quantity

Multiply the horizontal-axis units by the vertical-axis units.

\[\text{integral units}=\text{horizontal units}\times\text{vertical units}.\]

Context	Horizontal units	Vertical units	Integral meaning
Geometrical region	m	m	\(\mathrm{m^2}\): physical area
Earning rate	h	€/h	€: total earnings
Speed-time graph	h	km/h	km: distance when speed is non-negative
Flow rate	min	L/min	L: total volume

Interpret the units, not just the picture

Do not automatically write “square units”. Determine the product of the axis units and then explain what that accumulated quantity means.

Worked example 6: interpreting an integral in context

Water enters a tank at \(r(t)=2t+5\) litres per minute for \(0\le t\le6\). Find the total volume entering the tank during the first 6 minutes.

Solution

\[V=\int_0^6(2t+5)\,dt=66.\]

The units are \((\mathrm{L/min})(\mathrm{min})=\mathrm L\).

Answer The total volume is 66 litres.

Flow-rate graph with shaded accumulated volume — The numerical area under the rate graph represents volume, not square units.

Putting the ideas together

Many AI SL questions link the ideas in this order:

\[\text{derivative}\longrightarrow\text{particular function}\longrightarrow\text{definite integral / area}.\]

Worked example 7: from a derivative to an area

Given \(f'(x)=6x+4\) and \(f(0)=3\), find \(f(x)\), then find the area between \(y=f(x)\) and the x-axis from \(x=0\) to \(x=2\).

Solution

Anti-differentiate:

\[f(x)=\int(6x+4)\,dx=3x^2+4x+C.\]

Using \(f(0)=3\) gives \(C=3\), so \(f(x)=3x^2+4x+3\). This is positive on \([0,2]\), hence

\[A=\int_0^2(3x^2+4x+3)\,dx=22.\]

Answer \(f(x)=3x^2+4x+3\) and \(A=22\) square units.

TI-Nspire graph showing shaded area under y equals 3x squared plus 4x plus 3 from 0 to 2 — The GDC verifies the positive area of 22.

Practice questions

Questions 1–4: anti-differentiation and definite integrals

Find \(\int(8x^3-6x+5)\,dx\).
Given \(\frac{dy}{dx}=4x-3\) and \(y=7\) when \(x=2\), find \(y\) in terms of \(x\).
Given \(f'(x)=12x^2-2x\) and \(f(1)=5\), find \(f(x)\).
Use technology to evaluate \(\int_0^5(3x+4)\,dx\).

Questions 5–9: area and interpretation

Find the area enclosed by \(y=-x^2+6x\) and the x-axis.
Given \(f'(x)=2x+1\) and \(f(0)=2\), find \(f(x)\), then find the area between \(y=f(x)\) and the x-axis from \(x=0\) to \(x=3\).
Water flows into a tank at \(q(t)=3t+8\) litres per minute for \(0\le t\le5\). Write and evaluate a definite integral for the total volume. State the units.
A particle has velocity \(v(t)=t^2+2t+3\) for \(0\le t\le4\). Write a definite integral for its displacement and use technology to find the value.
IB-style. \(\frac{dy}{dx}=12-2x\), and the curve passes through \((0,5)\).
(a) Find \(y\) in terms of \(x\).
(b) Write a definite integral for the area between the curve and the x-axis from \(0\) to \(2\).
(c) Use technology to find the area.

Answer key

Answers 1–4

\(\int(8x^3-6x+5)\,dx=2x^4-3x^2+5x+C\).
\(y=2x^2-3x+C\). Using \(y(2)=7\) gives \(C=5\), so \(\boxed{y=2x^2-3x+5}\).
\(f(x)=4x^3-x^2+C\). Since \(f(1)=5\), \(C=2\), so \(\boxed{f(x)=4x^3-x^2+2}\).
\(\int_0^5(3x+4)\,dx=57.5\).

Answers 5–9

The intercepts are 0 and 6, and the curve is positive between them: \(A=\int_0^6(-x^2+6x)\,dx=36\) square units.
\(f(x)=x^2+x+2\). Then \(A=\int_0^3(x^2+x+2)\,dx=16.5\).
\(\int_0^5(3t+8)\,dt=77.5\). Units: litres.
\(\int_0^4(t^2+2t+3)\,dt\approx45.333\).
(a) \(y=12x-x^2+5\). (b) \(A=\int_0^2(12x-x^2+5)\,dx\). (c) \(A\approx28.667\).

Summary checklist

AI SL 5.5 checklist

If you see a derivative and want the original function, anti-differentiate.
For an indefinite integral, include \(+C\).
Use a point or value as a boundary condition to determine \(C\).
For a definite integral, write the integrand, variable and limits correctly before using the GDC.
Use technology to evaluate definite integrals in AI SL.
If the curve stays above the x-axis, area is \(\int_a^b f(x)\,dx\).
In context, interpret the integral as an accumulated quantity instead of automatically writing “square units”.
State the interval and contextual units clearly.

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