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?

math expression

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:

math expression

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:

LIVEMath


Now for the normal answer:

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.


Answer

There must be a better way! Integration was developed by Newton and Leibniz to save all this "adding areas of rectangles" work.

 

General Case

math expression

[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:

math expression

How do we evaluate this expression?

If F(x) is the integral of f(x), then

math expression

This means:

To evaluate a definite integral, follow these steps:

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:

math expression

 

Example 2:

Returning to our arches problem above...

math expression

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.


Answer

Mini-Lecture

See the
mini-lecture on the area under a curve.


Example 3:

Find the (exact) area under the curve y = x2 + 1 between x = 0 and x = 4 and the x-axis.


Here is the solution in LiveMath:

LIVEMath


Now for the conventional answer:

Answer



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