2. Area Under a Curve

by M. Bourne

We met areas under curves earlier in the Integration section (see 3. Area Under A Curve), but here we develop the concept further. (You may also be interested in Archimedes and the area of a parabolic segment, where we learn that Archimedes understood the ideas behind calculus, 2000 years before Newton and Leibniz did!)

It is important to sketch the situation before you start.

We wish to find the area under the curve y = f(x) from x = a to x = b.

We can have several situations:

Case 1: Curves which are entirely above the x-axis.

In this case, we find the area by simply finding the integral:

Where did this formula come from?

Area Under a Curve from First Principles

In the diagram, a "typical rectangle" is shown with width Δx and height y. Its area is yΔx.

If we add all these typical rectangles, starting from a and finishing at b, the area is approximately:

Now if we let Δx → 0, we can find the exact area by integration:

Example of Case 1:

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Find the area underneath the curve y = x² + 2 from x = 1 to x = 2.

Answer

Case 2: Curves which are entirely below the x-axis

(for the range of x values being considered):

mathImage

In this case, the integral gives a negative number. We need to take the absolute value of this to find our area:

Example of Case 2:

Find the area bounded by y = x² − 4, the x-axis and the lines x = -1 and x = 2.

Answer

Case 3: Part of the curve is below the x-axis and part of the curve is above the x-axis.

mathImage

In this case, we have to sum the individual parts, taking the absolute value for the section where the curve is below the x-axis (from x = a to x = c).

Example of Case 3:

What is the area bounded by the curve y = x³, x = -2 and x = 1?

Answer: We can see from the graph that the portion between x = -2 and x = 0 is below the x-axis, so we need to take the absolute value for that portion.

NOTE: In each of Case (1), (2) and (3), the curves are easy to deal with by summing elements L to R:

We are (effectively) finding the area by horizontally adding the areas of the rectangles, width dx and heights y (which we find by substituting values of x into f(x)).

(with absolute value signs where necessary).

Case 4: Certain curves are much easier to sum vertically

(or only possible to sum vertically).

In this case, we find the area is the sum of the rectangles, heights x = f(y) and width dy.

If we are given y = f(x), then we need to re-express this as x = f(y) and we need to sum from bottom to top.

So, in case 4 we have:

Example of Case 4:

Find the area of the region bounded by the curve

the y-axis and the lines y = 1 and y = 5.

Answer: Sketch first:

In this case, we express x as a function of y:

So the area is given by:

Note: For this particular example, we could have also summed it horizontally (integrating y and using dx).

1. Applications of the Indefinite Integral

3. Area Between 2 Curves

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