Evaluate definite integrals and apply integration to find plane areas between curves and lines

What is a Definite Integral?

Think of a definite integral as a way to measure the area under a curve between two points on the x‑axis. Imagine you have a garden bed shaped like a curve – the integral tells you how many square metres of soil you need to fill it.

Mathematically, for a function f(x) between a and b, we write:

$$\int_{a}^{b} f(x)\,dx$$

Here a is the left bound, b the right bound, and the integral gives the signed area (positive if f(x) is above the x‑axis).

Basic Rules for Evaluation

• Linearity: $$\int_{a}^{b} [c\,f(x) + d\,g(x)]\,dx = c\int_{a}^{b} f(x)\,dx + d\int_{a}^{b} g(x)\,dx$$
• Constant factor: $$\int_{a}^{b} c\,f(x)\,dx = c\int_{a}^{b} f(x)\,dx$$
• Sum/difference: $$\int_{a}^{b} [f(x) \pm g(x)]\,dx = \int_{a}^{b} f(x)\,dx \pm \int_{a}^{b} g(x)\,dx$$

Fundamental Theorem of Calculus

If F(x) is an antiderivative of f(x) (i.e. F'(x)=f(x)), then:

$$\int_{a}^{b} f(x)\,dx = F(b) - F(a)$$

So to evaluate an integral, find F(x), plug in the upper bound b and subtract the value at the lower bound a.

Example 1: Simple Polynomial Integral

Evaluate ∫₀³ (2x + 1) dx.

1. Find an antiderivative: F(x) = x² + x.
2. Apply the theorem: F(3) - F(0) = (9 + 3) - (0 + 0) = 12.

Answer: 12 square units.

Finding Plane Areas Between Curves

When two curves f(x) and g(x) cross, the area between them from a to b is:

$$A = \int_{a}^{b} \bigl|f(x) - g(x)\bigr|\,dx$$

If you know f(x) ≥ g(x) on the interval, the absolute value can be dropped:

$$A = \int_{a}^{b} [f(x) - g(x)]\,dx$$

Analogy: Imagine two rivers flowing side by side; the area between them is like the land that lies between the banks.

Example 2: Area Between y = x² and y = 2x + 3

1. Find intersection points: Solve x² = 2x + 3 → x² - 2x - 3 = 0 → (x-3)(x+1)=0 → x = -1, 3.
2. Choose interval where f(x) ≥ g(x) (here, f(x)=2x+3 is above g(x)=x² between -1 and 3).
3. Set up integral: A = ∫_{-1}^{3} [(2x+3) - x²] dx.
4. Find antiderivative: F(x) = x³/3 + 3x²/2 - x³/3 = 3x²/2 - x³/3 (simplify if needed).
5. Evaluate: F(3) - F(-1) = [27/2 - 27/3] - [1/2 + 1/3] = 9 - 5/6 = 49/6 ≈ 8.17.

Answer: ≈ 8.17 square units.

Quick Reference Table

Exam Tips & Tricks

• ?? Check the bounds: Make sure you plug in the correct upper and lower limits.
• ?? Units matter: If the function represents a rate (e.g., speed), the integral gives distance with appropriate units.
• ?? Sign of area: If the integrand is negative over an interval, the integral will be negative. Use absolute value if the question asks for area.
• ?? Sketch the curves: A quick sketch helps you decide which function is on top.
• ?? Look for symmetry: If the function is even or odd, you can simplify the integral.
• ?? Check your work: A quick sanity check (e.g., area should be positive) can catch sign errors.

Good luck, and remember: practice makes the integral easy! 🚀