SL 5.9 · Use definite integrals and numerical methods to find accumulated change and area

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Calculus

IB Mathematics: Analysis and Approaches SL — Study topic SL 5.9

What you will learn

• Interpret a definite integral as signed accumulated change over an interval.
• Evaluate definite integrals using an antiderivative when one is available.
• Distinguish signed area from total geometric area.
• Estimate accumulation from tabulated data using the trapezoidal rule.
• Use graphing technology to check an integral or numerical estimate while showing the mathematical reasoning.

1. From rate to accumulated change

• A definite integral represents net accumulated change when its integrand is a rate.
• Subtract the antiderivative value at the lower endpoint from its value at the upper endpoint.
• The integral’s units combine the units of the integrand and the input.

2. Reading area from a graph

• Area above the axis contributes positively to an integral; area below contributes negatively.
• Total geometric area is found by adding positive pieces.
• Check where graphs cross before calculating area.

3. Estimating accumulation from data

• The trapezoidal rule approximates the graph by straight segments between measured points.
• Keep intermediate values unrounded until the final answer.
• Call a result an estimate when it comes from sampled data, and give sensible accuracy.

4. Technology and a reliable checking routine

• Use a graph to identify sign changes and confirm the interval.
• Technology checks arithmetic; it does not replace explaining the method.
• Match the result to the question’s meaning and requested accuracy.

Interpreting an integral

Worked example

Exact accumulated change from a rate

1. Set up the accumulation
   Net change is the definite integral of the rate over the stated time interval. The result is in units of the quantity because the rate is in units per second and time is in seconds.
   $$\int_0^2 (3t^2-2t)\,dt$$
2. Find an antiderivative
   An antiderivative of $3t^2-2t$ is $t^3-t^2$: differentiating this expression gives the original rate.
   $$\int (3t^2-2t)\,dt=t^3-t^2$$
3. Evaluate at the endpoints
   Subtract the antiderivative value at $t=0$ from its value at $t=2$.
   $$[t^3-t^2]_0^2=(8-4)-(0-0)=4$$

Worked example

Total area when the graph crosses the axis

1. Locate the crossings
   Set the function equal to zero. The roots divide the interval into sections where the graph is above or below the axis.
   $$x^2-1=0\quad\Rightarrow\quad x=-1,\ 1$$
2. Identify the signs
   The graph is above the axis on $[-2,-1]$ and $[1,2]$, and below it on $[-1,1]$. By symmetry, the two outer areas are equal. Calculate the middle area using $1-x^2$ so it is positive.
   $$A=2\int_1^2(x^2-1)\,dx+\int_{-1}^1(1-x^2)\,dx$$
3. Evaluate and add positive areas
   An antiderivative of $x^2-1$ is $x^3/3-x$, and one of $1-x^2$ is $x-x^3/3$. The outer area on one side is $4/3$, so the two outer areas total $8/3$. The middle area is $4/3$. Add these positive areas.
   $$A=2\left[\frac{x^3}{3}-x\right]_1^2+\left[x-\frac{x^3}{3}\right]_{-1}^1=4$$

Worked example

Estimate accumulation from measured rates

1. Determine the interval width
   The measurements are equally spaced, so each interval has width $2$ minutes. Each trapezium’s area is its width multiplied by the average of its endpoint rates.
   $$h=2$$
2. Apply the trapezoidal rule
   The first and last rates receive half weight, and the two interior rates receive full weight. Multiplication by the time width gives an amount in litres.
   $$\frac{2}{2}[5+2(7)+2(6)+2]=33$$
3. Interpret the estimate
   The estimate assumes that the rate changes linearly between consecutive measurements. The units are litres per minute multiplied by minutes.
   $$33\text{ litres}$$

Common mistakes and how to avoid them

Lesson summary

• A definite integral of a rate gives net accumulated change over an interval.
• Evaluate an integral by subtracting the antiderivative’s lower-end value from its upper-end value.
• Signed area can involve cancellation; total area requires adding positive pieces.
• The trapezoidal rule estimates accumulation by joining neighbouring data points with straight segments.
• Use graphs and technology to check signs and arithmetic, while showing the setup, units, and interpretation.

Check your understanding

Key terms

Accumulated change

The net change in a quantity over an interval, found by adding the contributions of its rate.

Definite integral

A signed total over a specified interval, written with lower and upper endpoints.

Antiderivative

A function whose derivative is the given function.

Signed area

Area above the horizontal axis counted positively and area below it counted negatively.

Trapezoidal rule

A numerical method that estimates an integral by approximating sections of a graph with trapezia.

Continue through IB AA SL

View the complete IB AA SL International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL curriculum and lessons

• SL 5.1 · Interpret limits and derivatives as gradients and rates of change
• SL 5.2 · Connect derivative functions to increasing and decreasing behaviour
• SL 5.3 · Differentiate powers, trigonometric, exponential, and logarithmic functions
• SL 5.4 · Apply chain, product, and quotient rules
• SL 5.5 · Use second derivatives and interpret concavity
• SL 5.6 · Find and classify stationary points

About this lesson and its review

Published by DoAssignment. This AI-assisted lesson follows International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL, study topic SL 5.9. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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