In this lessons, we will learn how to use integrals (or integration) to find the areas under the curves defined by the graphs of functions. We also learn how to use integrals to find areas between the graphs of two functions.

We have also included
calculators and tools that can help you calculate the area under a curve and area between rwo curves.

Related Topics: More Calculus Topics

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* = *x*^{2} + 1, bounded below by *y* = *x*, and bounded on the sides by *x* = 0 and *x* = 1.

**Solution:**

The upper boundary curve is *y* = *x*^{2 }+ 1 and the lower boundary curve is *y* = *x*.

Using the formula,

**Example:**

Find the area between the two curves *y* = *x*^{2} and *y* = 2*x* – *x*^{2}.

**Solution:**

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

*x*^{2} = 2*x* – *x*^{2
}2*x*^{2} – 2*x* = 0

2*x*(*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

The Area Under a Curve: approx. the definite integral

Area under a curve using integration, step by step, example

Integration is used to determine the area under a curve. A parabola is drawn such that it intersects the x-axis. The x-intercepts are determined so that the area can be calculated.

Introduction - Area under a curve - definite integral

Definite Integrals and Area Under a Curve

Use the following Definite Integral Calculator to find the Area under a curve.

Enter the function, lower bound and upper bound.

Area under a curve using integration, step by step, example.

Finding Areas Between Curves

Area Between Curves - Integrating with Respect to y

Use the following Area between Curves Calculator to show you the steps and to check your answers.

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