Definite Integral (Area)
The net area between a curve and the x-axis from a to b.
Formula
Variables
Example
∫₀³ x dx equals the area of a triangle: ½·3·3 = 4.5.
How it works
A definite integral adds up infinitely many tiny slices of the graph between x = a and x = b. Slices above the x-axis count positively, while slices below the x-axis count negatively, which is why the result is the net or signed area rather than the total geometric area. This comes from approximating the region with thin rectangles and taking the limit as their width shrinks to zero.
When to use it
Use this when you need the accumulated effect of a rate over an interval, such as distance from velocity, work from force, or total growth from a rate of change. It is also the standard way to find net area between a curve and the x-axis on a graph.
Worked example, step by step
Find the signed area under f(x) = x^2 - 1 from x = 0 to x = 2. Step 1: Set up the integral: ∫_0^2 (x^2 - 1) dx Step 2: Find an antiderivative: ∫(x^2 - 1) dx = x^3/3 - x Step 3: Evaluate at the bounds: [x^3/3 - x]_0^2 Step 4: Substitute x = 2: 2^3/3 - 2 = 8/3 - 2 = 2/3 Step 5: Substitute x = 0: 0^3/3 - 0 = 0 Step 6: Subtract: 2/3 - 0 = 2/3 Final answer: ∫_0^2 (x^2 - 1) dx = 2/3
Common mistakes to avoid
- Forgetting that areas below the x-axis contribute negative values
- Using the antiderivative but not applying the upper limit minus the lower limit correctly
- Treating a definite integral as always giving total area instead of signed area
Did You Know?
A definite integral is a signed area — regions below the axis count as negative.
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