When a curve is described by parametric equations, say x=f(t) and y = g(t), then the area under this curve between given limits may be found by integration.
The following applet approximates the area bounded by the parametric curve defined by x=x(t) and y=y(t) for a ≤ t ≤ b. Simply enter the functions x(t) and y=y(t) and the values a, b (in radians) and 0 ≤ n ≤ 1,000, the number of subintervals.
The values a and b can be changed by simply typing a new value, such as "1.2345", "pi/2", "sqrt(5)+cos(3)", etc. You may also change these values by using the up/down arrow keys or dragging the corresponding point left or right. To move the center of the graph, simply drag any point to a new location. To label the x-axis in radians (i.e. multiples of \pi), click on the graph and press "control-r". To switch back, simply press "control-r" again.

## Software/Applets used on this page

This page uses an applet from David Little (Lecturer in the Mathematics Department, Penn State University, USA) and is used with his permission.

## Glossary

### graph

A diagram showing a relationship between two variables.

The diagram shows a vertical y axis and a horizontal x axis.

The diagram shows a vertical y axis and a horizontal x axis.

### integration

the process of finding an integral, the reverse process to differentiation.

### parametric equations

a pair of equations x=u(t) and y=v(t) where t is a parameter, that describe a curve.

## This question appears in the following syllabi:

Syllabus | Module | Section | Topic | Exam Year |
---|---|---|---|---|

AP Calculus BC (USA) | 4 | Integration | Parametric integration | - |

AQA A-Level (UK - Pre-2017) | C4 | Integration | Parametric integration | - |

CCEA A-Level (NI) | C4 | Integration | Parametric integration | - |

Edexcel A-Level (UK - Pre-2017) | C4 | Integration | Parametric integration | - |

Methods (UK) | M9 | Integration | Parametric integration | - |

OCR A-Level (UK - Pre-2017) | C4 | Integration | Parametric integration | - |

OCR-MEI A-Level (UK - Pre-2017) | C4 | Integration | Parametric integration | - |

Pre-U A-Level (UK) | 5 | Integration | Parametric integration | - |

Universal (all site questions) | I | Integration | Parametric integration | - |

WJEC A-Level (Wales) | C4 | Integration | Parametric integration | - |