Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0000 362024 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0864 1211280 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0864 1211280 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1555 4191992 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1569 4219528 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1576 4237240 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1578 4267760 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1589 4290480 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1589 4290480 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2115 4968168 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2151 5160320 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2288 5181248 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.2399 5378040 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.2399 5378296 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

The Trapezium Rule

The trapezium rule is a numerical method for estimating integrals. It is most useful when there is no analytical answer to an integral, and only a number is needed. It works by approximating the area under the curve by a series of trapezia, then evaluating the areas and adding them up.

The area under the curve is approximated by a series of trapezia.

The formula for the integral rule iswhereis the step size or the increment by which the values ofincrease, 1 in the diagram above andis defined byeg

For ease of calculation it is a good idea to tabulate the values of the

0

1

2

3

4

5

6

0

1

2

3

4

5

6

0

5

8

9

8

5

0

This estimate is an underestimate for the integral. The curve is concave – it curves down. In general the trapezium method gives an underestimate for the integral of concave functions and an overestimate for convex functions, an example of which is given below.