Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Calculus — Topic 5.5
📋 Learning Intentions
- Calculate definite integrals using the GDC and interpret the result in context.
- Find the area enclosed between a curve and the x-axis, distinguishing between signed and unsigned area.
- Identify when a definite integral gives a negative value and explain what this means geometrically.
- Apply the correct method when a curve crosses the x-axis within the interval of integration.
- Interpret definite integrals in real-world contexts such as displacement, accumulated change, and area.
- Use the TI-84 Plus CE to evaluate definite integrals numerically with correct syntax and accuracy.
Real-World Introduction
A solar panel installation company wants to calculate the total energy generated over a six-hour period. The power output (in kilowatts) varies throughout the day and can be modelled by a function of time. To find the total energy, the company must accumulate the power over the time interval — this is precisely what a definite integral computes.
In environmental science, the area under a concentration-time graph gives the total exposure to a pollutant. In economics, the area under a marginal cost curve gives total cost. In each case, the definite integral transforms a rate into a total accumulated quantity — one of the most powerful ideas in applied mathematics.
🌍 Why This Matters The definite integral is the mathematical engine behind calculating total distance from velocity, total revenue from marginal revenue, and total drug concentration in pharmacokinetics. Every time a quantity accumulates over time or space, a definite integral is the tool that computes it.
Core Concepts & Definitions
Differentiation answers the question: "how fast is this changing?" Integration answers the reverse question: "given the rate of change, what is the total accumulated amount?" The definite integral takes this idea and restricts it to a specific interval \([a, b]\), producing a single numerical value rather than a general function.
📖 Key Definition — The Definite Integral The definite integral of a continuous function \( f(x) \) over the interval \([a, b]\) is defined as: \[ \int_{a}^{b} f(x)\, dx = F(b) - F(a) \] where \( F(x) \) is any antiderivative of \( f(x) \). The values \( a \) and \( b \) are called the lower limit and upper limit of integration respectively. The result is