01 Key Concepts
Area Under a Single Curve
For f(x) >= 0 on [a,b], the area under the curve equals Integral from a to b of f(x) dx.
Area Below the x-axis
If f(x) < 0, the definite integral gives a negative value; take the absolute value to get the actual area.
Area Between Two Curves
If f(x) >= g(x) on [a,b], the area between them is Integral from a to b of [f(x) - g(x)] dx.
Finding Intersection Points First
Before computing area between two curves, solve f(x) = g(x) to find the correct bounds of integration.
Riemann Sums (Conceptual Basis)
Area under a curve can be approximated by summing the areas of many thin rectangles — the definite integral is the exact limit of this process.
02 Key Formulas
- Area = Integral from a to b of f(x) dx (when f(x) >= 0)
- Area between curves = Integral from a to b of [f(x) - g(x)] dx
03 Solved Examples
Example 1 Find the area under f(x) = x^2 from x=0 to x=3.
- Area = Integral from 0 to 3 of x^2 dx.
- Antiderivative: x^3/3.
- Evaluate: 27/3 - 0.
Answer: 9 square units
Example 2 Find the area between f(x) = x+2 and g(x) = x^2 from x=0 to x=1.
- Since f(x) >= g(x) on this interval, Area = Integral from 0 to 1 of [(x+2) - x^2] dx.
- Antiderivative: x^2/2 + 2x - x^3/3.
- Evaluate at x=1: 1/2 + 2 - 1/3 = 13/6. Evaluate at x=0: 0.
Answer: 13/6 square units
Example 3 Find the total area between f(x) = x^3 and the x-axis from x=-1 to x=1.
- Since x^3 is negative on [-1,0] and positive on [0,1], integrate each piece and take absolute values.
- Integral from -1 to 0 of x^3 dx = -1/4, so its area contribution is |−1/4| = 1/4.
- Integral from 0 to 1 of x^3 dx = 1/4.
- Total area = 1/4 + 1/4.
Answer: 1/2 square unit
04 Practice Questions
1Find the area under f(x)=x from x=0 to x=4.
8
2Find the area under f(x)=2 from x=1 to x=5.
8
3Find the area under f(x)=x^2 from x=0 to x=2.
8/3
4Find the area between f(x)=4 and g(x)=x^2 from x=-2 to x=2.
32/3
5If a definite integral gives -6 for a region below the x-axis, what is the actual area?
6
📄 Area Under Curve — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.