Area under simple curves

Area Under Simple Curves An area under a curve is the region bounded by a curve, its base, and its height. It represents the total area of the region. L...

Area Under Simple Curves#

An area under a curve is the region bounded by a curve, its base, and its height. It represents the total area of the region.

Let's consider a simple curve, like the one pictured below:

[Image of a simple curve]

The area under this curve represents the total area of the region bounded by the curve and the x-axis.

To find the area, we need to integrate the function representing the curve with respect to x. This means adding up the areas of infinitely small rectangles under the curve.

Formal Definition:

The area under a curve f(x) in the domain [a, b] is given by the integral:

$\int_a^b f(x) dx$

In this definition:

dx is the increment in x.
a is the starting value of x.
b is the ending value of x.
f(x) is the function representing the curve.

Interpreting the Area:

The area represents the total amount of the region under the curve between the x-axis and the curve. It tells us the total area of the shaded region in the graph.

Examples:

Area of a triangle: If the height and base of a triangle are represented by functions f(x) and g(x) respectively, then the area is given by the integral:

$\int_0^b f(x) - g(x) dx$

Area of a semicircle: If the radius of a semicircle is represented by a function, then the area is given by the integral:

$\pi \int_0^r r^2 dx$

Area of a parabola: If the equation of a parabola is f(x) = x^2, then the area is given by the integral:

$\int_0^r x^2 dx$

By understanding the concept of areas under curves, we can find the total area of various shapes by integrating appropriate functions with respect to x