Interpreting area and gradients
Gradients & areas under a curve
For a curved graph, the gradient is constantly
changing.
In order to find the gradient of a particular point
on a curved graph, a tangent needs to be drawn
at that point.
The gradient of the tangent drawn is then
worked out.
For example:
y
20
Tangent
1
+
Area under the
graph represents the
distance travelled
1
y3
1
Time
x
Example
Here is the velocity-time graph of a skier:
h
3
4
5
6
x
Change in x
1 Draw the tangent so that it just touches the
curve at that point.
2 Choose two points on the tangent and draw
a triangle.
h
h
h
x
h
For the tangent shown above:
change in y
19 – 1
18
Gradient =
=
=
= 5.806
change in x 4.8 – 1.7 3.1
0
y
y = x2
(5, 25)
25
1
= 100 + 225 + 275 + 250 + 100
= 950m
0
20
40
60
100 x
80
Time (s)
Acceleration is the gradient of the curve. So
draw the tangent at 20s, then find its gradient.
(4, 16)
15
y
10
15
(3, 9)
Velocity (m/s)
4 Decide whether the gradient is uphill (+) or
downhill (–).
(1, 1)
0
1
4
10
(2, 4)
5
0
1
[ 2 × 20 × 10]
a) Calculate an estimate for the acceleration of
the skier 20 seconds after the start.
20
3 Find the gradient of the triangle (do not count
the squares as the scales may be different).
change in y
Gradient =
change in x
+
+
5
Example
Find the area under the graph y = x2 for the
values of x between 0 and 5.
2
3
4
5
Using the trapezium rule gives:
6
20
5
x
0
20
1
1
= 2 (0 + 2(30) + 25) =
1
2
Area = 42.5 squared units
× (85)
Gradient =
4
20
D
C
B
A
0
h =1
A = 2 (0 + 2(1 + 4 + 9 + 16) + 25)
1
[ 2 × 20 × (12 2 + 15)]
[ 2 × 20 × (15 + 10]
10
y5
y4
1
[ 2 × 20 × (10 + 12 2 )]
+
Velocity (m/s)
y2
y1
5
2
Distance = A + B + C + D + E
= [ 2 × 20 × 10]
15
y0
1
Acceleration
gradient of tangent
This is the area under the graph. Use triangles
and trapeziums to estimate this area.
y
Change in y
0
y
y
Tangent just
touches at x = 3
10
0
The area under the graph represents the total
distance travelled.
b) Estimate the total distance travelled by the
skier.
40
60
Time (s)
= 0.2 ms-1
E
80
100 x
Care needs to be taken
when finding the gradient
since the scales on both
axes are different.
Algebra
Algebra
h
(y + 2(y1 + y2 + y3 + ...yn – 1) + yn)
2 0
where h is the width of the trapeziums and y1,
y2 ... yn are the lengths of the parallel sides of
the trapeziums. (Note: the trapeziums must be
of equal width).
y = x2
Curve
15
For a curved graph, you can find an approximate
area by splitting up the graph into trapeziums.
The area of each trapezium is then calculated
and then added together. Another way is to use
the trapezium rule, which is given by
Area =
25
Velocity-time graphs often appear on the exam
paper. The acceleration or deceleration can be
found by drawing a tangent at that point.
Area under a curve
– the trapezium rule
Velocity
Tangents and gradients
The gradient represents the rate of change. The
area under a graph is the total amount.