0A building has parabolic archways and we need to supply glass to close in the archways.
How much glass is needed?
To answer this, we need to know the area under the curve.
We'll see how to do this in 2 ways on this page:
- Using an approximation (finding areas of rectangles).
- Using integration.
Before integration was developed, mathematicians could only approximate the answer by dividing the space into rectangles and adding the areas of those rectangles, something like this:
The height of each rectangle is found by calculating the function values, as shown for the typical case x = c, where the rectangle height is f(c). We get a better result if we take more and more rectangles.
In the above diagram, we are approxcimating the area using inner rectangles (each rectangle is inside the curve). We could also find the area using the outer rectangles.
