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: 0.0001 360704 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0972 1210904 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0972 1210904 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1723 4044040 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1737 4071728 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1746 4089440 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1746 4106448 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1755 4114184 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1755 4114184 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.3227 4749528 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.3254 4834760 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.3260 4855688 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.3380 5019688 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.3381 5019944 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Integrating When a Curve is Given in Parametric Form

The area under a curvebetween the limitsandis given by the integralIf however,andare given as functions of a parametera lot more work may be required. Instead of integrating with respect towe may integrate with respect to t usinghence

Example: A curve is given by the parametric equationsFind the area under the curve between the values

Example: A curve is given by the parametric equationsFind the area under the curve between the values

We expand the brackets to obtainThe integral becomes

To evaluate this integral we rearrange the identityto give

This method has the advantage of making the integral of many closed curves much simpler, since they may often be parametrized in terms of a parameter that varies between 0 and

Add comment

Security code
Refresh