You are getting a routing error because Rails is looking for the image sources in the wrong directory.
Try this:
<div class="carousel-inner">
<div class="item active">
<%= image_tag "carousel/1", :alt => "1" %>
</div>
<div class="item">
<%= image_tag "carousel/2", :alt => "2" %>
</div>
<div class="item">
<%= image_tag "carousel/3", :alt => "3" %>
</div>
</div>
I'd consider doing this with a loop:
<div class="carousel-inner">
<% (1..3).each do |i| %>
<div class="item">
<%= image_tag "carousel/#{i.to_s}", :alt => i.to_s %>
</div>
<% end %>
You could set the index of the active item to some variable, and use that to set the active:
<div class="carousel-inner">
<% (1..3).each do |i| %>
<div class="item <%= 'active' if i == active_index %>">
<%= image_tag "carousel/#{i.to_s}", :alt => i.to_s %>
</div>
<% end %>