Why is Calculus Intimidating? What makes it different from the other math subjects that came before? You are still working with equations. You are still working with graphs and functions. You are still finding areas and slopes.
Here's one reason why: "In all your previous math classes, you only learned how to find areas and volumes of silly little figures, like squares and cubes and maybe you remember a nonagon or cone. But what about crazy, strange, weird areas? Volumes of crazy, strange, weird figures? Did you ever wonder where the formula for the volume of a sphere come from? Calculus not only can answer questions about position, velocity, and acceleration, and how to maximize and minimize quantities, but it can do all this other stuff too." (http://samjshah.com/2009/02/26/how-do-you-introduce-integrals/)
While it is great that Calculus can help us do these things, it sure is intimidating! When you get to the point where a student comes to this realization, here's one great way to diffuse the situation.
1) Have a piece of graph paper handy.
2) Draw an x and y axis. Number them.
3) Draw a squiggly line that crosses the x-axis a few times.
4) Pick out one area above the x-axis that is enclosed by the curve and the x-axis.
NOW INVOLVE THE STUDENT:
5) Ask the student to figure out what the area under the curve is.
6) Give the student a minute or so. Maybe they come up with an approach, maybe they don't.
7) If they decide to draw a series of columns using the graph paper and start counting the boxes, say "Exactly! Good Job!"
Otherwise jump in and say "How about we draw some rectangles and start counting boxes?"
8) From there you arrive at an approximate area. At that point you say "Congratulations, you just learned how to do an integral!"
Of course it's not that simple, but you go on to explain, "Finding an integral is just like finding an area under a curve. That's it. Not so bad!"
CONGRATULATIONS! YOU HAVE JUST MADE CALCULUS SEEM EASY!
Return to this concept as many times as you need to when integrals get frustrating to remind the student in a visual way what calculus is doing.