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?

Area under parabolic arches

We need to know the area under the curve.

We'll see howto do this in 2 ways on this page:

  1. Using an approximation (finding areas of rectangles)
  2. Using integration

Before integration was developed, they could only approximate the answer by dividing the space into rectangles and adding the areas of those rectangles:

Area under curves - rectangles

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.

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 rectangles method.

You can play with this concept further in the Reimann Sums section.

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.

General Case

Area under a curve by integration

[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:

`"Area"=int_a^bf(x)dx`

How do we evaluate this expression?

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

`int_a^bf(x)dx=[F(x)]_a^b=F(b)-F(a)`

This means:

To evaluate a definite integral, follow these steps:

This forms part of The Fundamental Theorem of Calculus.


Example 2: Evaluation of Definite Integral

Evaluate: `int_1^(10)3x^2dx`

Example 3: Arches problem

Returning to our arches problem above...

Area of parabolic window

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.

Example 4

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

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