In these lessons and solutions, 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 two curves.

Related Topics: More Calculus Topics

**Formula for Area bounded by curves (using definite integrals)**

**How to find the Area between Curves?**
### Videos - Definite Integral

The Area Under a Curve: approximate the definite integral

Example: Approximate the area under the curve f(x) = x^{2} (i.e. the area between y = x^{2} and y = 0) from x = 1 to 3. Use n = 4 rectangles.

How to find the area under a curve using integration, step by step, example.

Example: Find the area bounded by the curves y = x^{2} - 6x + 9 and y = x + 3.
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.

Example: Calculate the area enclosed by the curve y = 2x - x^{2} and the x-axis.

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.

Finding Areas Between Curves

Example: Find the area bounded by the curves y = x^{2} - 4x and y = 2x.
Area Between Curves - Integrating with Respect to y.

Example: Find the area bounded by the curves x = y^{3} - y and x = 1 - y^{4}.
Use the following Area between Curves Calculator to show you the steps and to check your answers.

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

We have also included calculators and tools that can help you calculate the area under a curve and area between two 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 for area bounded by curves,

**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

Example: Approximate the area under the curve f(x) = x

Example: Find the area bounded by the curves y = x

A parabola is drawn such that it intersects the x-axis. The x-intercepts are determined so that the area can be calculated.

Example: Calculate the area enclosed by the curve y = 2x - x

Introduction - Area under a curve - definite integral

Definite Integrals and Area Under a Curve

Enter the function, lower bound and upper bound.

Example: Find the area bounded by the curves y = x

Example: Find the area bounded by the curves x = y

You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

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