3. The Area under a Curve
by M. Bourne
A 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 howto 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.
[This method was known to the Ancient Greeks. See Archimedes and the area of a parabolic segment.]
See the Riemann Sums applet where you can interactively explore this concept.
Example 1: Approximation using rectangles
(a) Find the area under the curve y = 1 − x2 between x = 0.5 and x = 1, for n = 5, using the sum of areas of rectangles method.
(b) Find the area under the curve given in part (a), but this time use n = 10, using the sum of areas of (upper) rectangles method.
You can play with this concept further 0n the Reimann Sums page.
Finding Areas using Definite Integration
There must be a better way than finding areas of rectangles! Integration was developed by Newton and Leibniz to save all this "adding areas of rectangles" work.
The curve y = f(x), completely above x-axis. Shows a "typical" rectangle, Δx wide and y high.
[NOTE: The curve is completely ABOVE the x-axis].
When Δx becomes extremely small, the sum of the areas of the rectangles gets closer and closer to the area under the curve. If it actually goes to 0, we get the exact area.
We use integration to evaluate the area we are looking for. We can show in general, the exact area under a curve y = f(x) from `x = a` to `x = b` is given by the definite integral:
How do we evaluate this expression?
If F(x) is the integral of f(x), then
To evaluate a definite integral, follow these steps:
- integrate the given function (do not include the K)
- substitute the upper limit (b) into the integral
- substitute the lower limit (a) into the integral
- subtract the second value from the first value
- the answer will be a number
This forms part of The Fundamental Theorem of Calculus.
Example 2: Evaluation of Definite Integral
Example 3: Arches problem
Returning to our arches problem above...
If the arch is 2 m wide at the bottom and
is 3 m high,
(i) find the equation of the parabola
(ii) find the area under each arch using integration.
Find the (exact) area under the curve y = x2 + 1 between x = 0 and x = 4 and the x-axis.