4/8/2019 2.

Area Under a Curve by Integration

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2. Area Under a Curve by Integration

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.

f(x)
y = f(x)

x
a Δx b
The curve y = f(x), completely above the x-axis. Shows a "typical" rectangle, Δx wide and y high.

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

b
Area = ∫ f (x) dx
a

Where did this formula come from?

Area Under a Curve from First Principles

In the diagram above, 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:

b
∑(y )Δx
x=a

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

b
Area = ∫ f (x) dx
a

This follows from the Riemann Sums, from the Introduction to Integration chapter.

Example of Case 1
2
Find the area underneath the curve y = x + 2 from x = 1 to x = 2.

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Answer

y(x)
20

15

10

5
x
-3 -2 -1 1 2 3
2
The curve y = x + 2, showing the portion under the curve from x = 1 to x = 2.

b
Area = ∫ f (x) dx
a
2
= ∫ (x2 + 2) dx
1
2
x3
=[ + 2x]
3 1

8 1
= [( + 4) − ( + 2)]
3 3
13
= units2
3

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

We consider the case where the curve is below the x-axis for the range of x values being considered.

f(x)
a Δx b x

y = f(x)

The curve y = f(x), completely below the x-axis. Shows a "typical" rectangle, Δx wide and y high.

In this case, the integral gives a negative number. We need to take the absolute value of this to find our area:
∣ b ∣
∣ ∣
Area = ∣ ∫ f (x) dx ∣
∣ a ∣
∣ ∣
Example of Case 2

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

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Answer

y
4

2
x
-3 -2 -1 1 2 3
-2

-4

The curve y = x 2 − 4, showing the portion under the curve from x = −1 to x = 2.

The required area is totally below the x-axis in this example, so we need to use absolute value signs.

∣ b ∣
∣ ∣
Area = ∣ ∫ f (x) dx ∣
∣ a ∣
∣ ∣
∣ 2 ∣
= ∣∣ ∫ (x2 − 4) dx ∣∣
∣ −1 ∣
∣ 3 2 ∣
= ∣[ − 4x] ∣
∣ x ∣
∣ 3 ∣
−1 ∣
∣
∣ 8 1 ∣
= ∣∣ [( − 8) − (− + 4)]∣∣
∣ 3 3 ∣
= ∣−9∣

= 9 units2

Case 3: Part of the curve is below the x-axis, part of it is above the x-axis
f(x)

a x
c b

y = f(x)

From a to c, the curve y = f(x) is below the x-axis, and from c to b, it is above.

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).

∣ c ∣ b
Area = ∣ ∫ f (x) dx ∣ + ∫ f (x) dx
∣ ∣
∣ a ∣ c

If we don't do it like this, the "negative" area (the part below the x-axis) will be subtracted from the "positive" part, and our
total area will not be correct.

Example of Case 3
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3
What is the area bounded by the curve y = x , x = −2 and x = 1?

Answer

y
2
x
-2 -1 1
-2

-4

-6

-8

-10

The curve y = x 3, showing the portion under the curve from x = −2 to x = 1.

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.

∣ 0 ∣ 1
Area = ∣∣ ∫ x3 dx ∣∣ + ∫ x3 dx
∣ −2 ∣ 0

∣ 4 0 ∣ 1
x4
= ∣[ ] ∣ + [ ]
∣ x ∣
∣ 4 −2 ∣ 4 0
∣ ∣
∣ 16 ∣∣ 1
= ∣∣ (0 − )∣ + ( − 0)
∣ 4 ∣ 4
1
=4+
4
= 4.25 units2

Don't do it like this!

If you just blindly find the integral from the lower limit to the upper limit, you won't get the actual area in such cases.
1
Integral = ∫ x3 dx
−2
1
x4
=[ ]
4 −2

1 16
=( − )
4 4
15
=−
4
= −3.25
This is not the correct answer for the area under the curve. It is the value of the integral, but clearly an area cannot be
negative.

It's always best to sketch the curve before finding areas under curves.

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Summary (so far)

In each of Case 1, Case 2 and Case 3, we are summing elements left to right, like this:

f(x)
y = f(x)

y
x
a dx b
Area under a curve using vertical rectangles (summing left to right).

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)).

So

b
A = ∫ f (x) dx
a

(with absolute value signs where necessary, if the curve goes under the x-axis).

Case 4: Certain curves are much easier to sum vertically

In some cases, it is easier to find the area if we take vertical sums. Sometimes the only possible way is to sum vertically.

dy
x

c
x

The best way to find the area under this curve is by summing 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:

d
A = ∫ f (y ) dy
c

Example of Case 4

Find the area of the region bounded by the curve y = x − 1, the y -axis and the lines y = 1 and y = 5.

Answer

Sketch first:

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y
6

5
x = y2 + 1
4
dy
x
3

1
x
5 10 15 20 25 30

The curve x = y 2 + 1, showing the portion "under" the curve from y = 1 to y = 5.

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

y= x−1

y2 = x − 1

x = y2 + 1
So the area is given by:
5 5
y3
A = ∫ (y 2 + 1) dy = [ + y]
1 3 1

1
= 45 sq units
3
Note: For this particular example, we could have also summed it horizontally (integrating y and using dx ), but we would
need to break it up into sections first.

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