# Areas under curve and averages

johnnyamerica
This is a problem I'm working on by myself and I'm not sure how to go about it because I do not know how to phrase the question in calculus terms.

Imagine you have a simple quadrilateral between x=1 and x=4. A line passes from 1,2 to 4,5. The average height of the two points or of the line is 3.5, and the width of the shape is 3. The area when multiplying the average height by the width would be 10.5.

Now what if, instead of having a straight line from 1,2 to 4,5 there were a curve? I've taken the heights between the two closest points, averaged them and multiplied them by the width. I can only do this with two points (I can't take the average of three heights and multiply them by the width between the three points).

With a graph, I would have sets of two different heights at a time, take the average and multiply them by the width between, move on to the second & third point, third & fourth point, fourth & fifth and so on. I would then add up all the areas.

With points on a curve I'm not sure how to go about phrasing this because there would be an infinite number of different heights.

I'm just looking for any hints or suggestions right now. I'll continue to work on the problem but directing me to any similar problems will be very helpful.