Why would we want to approximate the integral of the texture value over the texture space regions, and how does this prevent aliasing more than trilinear filtering?

Why we have the rectangle shape for both rendered image and texture space? Should one of them be trapezoid?

yes. The texture space ones should be trapezoids.

An important takeaway from this slide: for the green box that represents a "far away" location in the image, the texture space is not sampled enough in the u direction as in the v direction (green box narrower along u), causing blurring in the trilinear filtering result.