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.

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:

Need Graph Paper?

Download graph paper

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).