Area Under A Curve

We can use integrals to find the areas under the curves defined by the graphs of functions. We can also use integrals to find areas between the graphs of two functions.

Formula for Area

The Area A of the region bounded by the curves y = f(x), y = g(x) and the lines x = a, x = b, where f and g are continuous f(x) >= g(x) for all x in [a, b} is

Example:

Find the area of the region bounded above by y = x² + 1, bounded below by y = x, and bounded on the sides by x = 0 and x = 1.

Solution:

The upper boundary curve is y = x² + 1 and the lower boundary curve is y = x.

Using the formula,

Example:

Find the area between the two curves y = x² and y = 2x – x².

Solution:

Step 1: Find the points of intersection of the two parabolas by solving the equations simultaneously.

x² = 2x – x²
2x² – 2x = 0
2x(x – 1) = 0
x = 0 or 1

The points of intersection are (0, 0) and (1, 1)

Step 2: Find the area between x = 0 and x = 1

Videos

The Area Under a Curve: approx. the definite integral

Finding Areas Between Curves

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