How can I specify that an area of my theme contains widgets?

You can add widgets conditionally by adding a conditional tag in your template file.

You also mentioned a image slider.

You can conditionally load the slider so it only displays where you want like the front page only.

An example would be to add this code in your header.php file so the slider only loads on the front page.

<?php if (function_exists('slider_template_tag') && is_front_page()) { slider_template_tag('82'); }?>

You can do the same with the widgets.

<?php if ( is_active_sidebar( 'left-sidebar' ) && is_page('007') ) : ?>
<ul id="sidebar">
    <?php dynamic_sidebar( 'left-sidebar' ); ?>
</ul>
<?php endif; ?>

This displays the widget area only on the page with i.d of 007.