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?
We need to know the area under the curve.
Before integration was developed, they could only approximate the answer by dividing the space into rectangles and adding the areas:
The height of each rectangle is found by calculating the function values, as shown for the typical case x = c. We get a better result if we take more and more rectangles.
Example of 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.
Here we can see a LiveMath solution with a graph animation:
Now for the normal answer:
(b) Find the area under the curve given in part (a), but this time use n = 10, using the sum of areas of rectangles method.
There must be a better way! Integration was developed by Newton and Leibniz to save all this "adding areas of rectangles" work.
General Case
[NOTE: The curve is completely ABOVE the x-axis].
By a Δ-process, we can show that 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
This means:
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.
Mini-Lecture
See the mini-lecture on the difference between definite and indefinite integrals.
Example 1: Evaluation of Definite Integral:
Example 2:
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.
Mini-Lecture
Example 3:
Find the (exact) area under the curve y = x2 + 1 between x = 0 and x = 4 and the x-axis.
For comparison, here is the solution in LiveMath:
Find your integral using Mathematica!
Enter multiply using *, square root of x using Sqrt[x] and trigonometry like Sin[x]. See the full list of how to enter math.
Click the blue button to find your integral.
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