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Area Under Curves

Calculating the exact space beneath a graph using Calculus.

Finding the area of a square, rectangle, or triangle is easy—we have simple formulas for those. But how do you find the area of a shape with a curved boundary? This was a problem that baffled mathematicians for centuries until the invention of Integral Calculus.

1. The Concept: Approximating with Rectangles

Before we had the exact formula, mathematicians approximated the area by filling the space under the curve with rectangles. This is known as a Riemann Sum.

If you use 4 wide rectangles, the answer is rough. If you use 100 thin rectangles, it's better. If you use infinite rectangles of zero width, you get the exact area.

[Image of Riemann sum rectangles approximating area under curve]

2. The Exact Method: Definite Integrals

The "Area Problem" is solved using the Definite Integral. If we want the area between the function f(x) and the x-axis, from point a to point b, we calculate:

Area = ∫ₐᵇ f(x) dx

This is evaluated by finding the antiderivative F(x) and calculating F(b) - F(a).

[Image of definite integral area under curve graph]

3. Step-by-Step Example

Problem: Find the area under the curve y = x² from x = 0 to x = 3.

Step A: Set up the Integral

Area = ∫₀³ x² dx

Step B: Find the Antiderivative

Using the Reverse Power Rule (add 1 to power, divide by new power), the antiderivative of x² is x³/3.

Step C: Evaluate

Upper Limit (3): 3³ / 3 = 27 / 3 = 9
Lower Limit (0): 0³ / 3 = 0
Area = 9 - 0 = 9 square units.

4. Net Signed Area

One tricky aspect of integrals is that they measure "signed" area. Area above the x-axis is positive. Area below the x-axis is treated as negative.

If you integrate sin(x) from 0 to 2π, the answer is 0, because the "hill" (positive area) exactly cancels out the "valley" (negative area).

Note: If you need the total physical area (treating both parts as positive), you must integrate the absolute value: ∫ |f(x)| dx.

5. Area Between Two Curves

Often, we need to find the area sandwiched between two different functions, f(x) and g(x). The logic is simple: Take the area of the top curve and subtract the area of the bottom curve.

[Image of area between two curves graph]
Area = ∫ₐᵇ [ Top(x) - Bottom(x) ] dx

For example, to find the area between y = x (top) and y = x² (bottom) from 0 to 1:

∫₀¹ (x - x²) dx
= [x²/2 - x³/3] evaluated from 0 to 1
= (1/2 - 1/3) - 0
= 3/6 - 2/6 = 1/6