When you have to integrate a polynomial with more than 1 term, integrate each term. So:
Definite Integrals
In the above examples, there was always a constant term left over after integrating. For this reason, such integrals are known as indefinite integrals. With definite integrals, we integrate a function between 2 points, and so we can find the precise value of the integral and there is no need for any unknown constant terms.
Finding the area under a curve
The area under a curve can be found be integrating, if the equation of the curve is known.
To find the area under the curve y = f(x) between x = a and x = b, integrate y = f(x) between the limits of a and b.
Areas under the x-axis will come out negative and areas above the x-axis will be positive. This means that you have to be careful when finding an area which is partly above and partly below the x-axis.
The second derivative
The second derivative is what you get when you differentiate the derivative.
Stationary Points
The second derivative can be used as an easier way of determining the nature of stationary points.
A stationary point on a curve occurs when dy/dx = 0. Once you have estabished where there is a stationary point, the type of stationary point (maximum, minimum or point of inflexion) can be determined using the second derivative.
If d²y is positive, then it is a minimum point.
dx²
If d²y is negative, then it is a maximum point.
dx²
If d²y is zero, then it could be a max, a min or a point of inflexion.
dx²
If d²y/dx² = 0, you must test the values of dy/dx either side of the stationary point, as before.
Coordinate geometry
The distance between two points
The length of the line joining the points (x1, y1) and (x2, y2) is:
Example:
Find the distance between the points (5, 3) and (1, 4).
Distance = √ [(1 - 5)² + (4 - 3)²]
= √ [16 + 1] = Ö17
The midpoint of a line joining two points
The midpoint of the line joining the points (x1, y1) and (x2, y2) is:
[½(x1 + x2), ½(y1 + y2)]
Example:
Find the coordinates of the midpoint of the line joining (1, 2) and (3, 1).
[½(4), ½(3)] = (2, 1.5)
The gradient of a line joining two points
The gradient of the line joining the points (x1, y1) and (x2, y2) is:
y2 - y1
x2 - x1
Example:
Find the gradient of the line joining the points (5, 3) and (1, 4).
Gradient = 4 - 3 = 1 = -0.25
1 - 5 -4
The equation of a line using one point and the gradient
The equation of a line which has gradient m and which passes through the point (x1, y1) is:
y - y1 = m(x - x1)
Example:
Find the equation of the line with gradient 2 passing through (1, 4).
y - 4 = 2(x - 1)
y - 4 = 2x - 2
y = 2x + 2
Since m = y2 - y1
x2 - x1
The equation of a line passing through (x1, y1) and (x2, y2) is therefore:
y - y1 = y2 - y1
x - x1 x2 - x1
Sin, Cos, Tan
The sine, cosine and tangents of common angles:
These occur frequently and need to be remembered.
Quadrants and the 'cast' rule
On a set of axes, angles are measured anti-clockwise from the positive x-axis. So 30º would be drawn as follows:
The angles which lie between 0º and 90º are said to lie in the first quadrant. The angles between 90º and 180º are in the second quadrant, angles between 180º and 270º are in the third quadrant and angles between 270º and 360º are in the fourth quadrant:
In the first quadrant, the values for sin, cos and tan are positive.
In the second quadrant, the values for sin are positive only.
In the third quadrant, the values for tan are positive only.
In the fourth quadrant, the values for cos are positive only.
This can be summed up as follows:
In the fourth quadrant, Cos is positive, in the first, All are positive, in the second, Sin is positive and in the fourth quadrant, Tan is positive. This is easy to remember, since it spells 'cast'.
Related angles
The sines, cosines and tangents of some angles are equal to the sines, cosines and tangents of other angles. For example, cos(-30º) = cos(30º) and cos(30º) = cos(390º) . In the following diagrams, the sines, cosines and tangents of each of the shaded angles have the same magnitude (ø is the same angle in each diagram):
For example, if ø is 30º,
sin30º = 0.5
sin150º = 0.5
sin210º = -0.5
sin330º = -0.5
These angles are 'related angles' and their cosines and tangents will be related in a similar way. Note that the signs of the sines (/cosines/tangents) are found using the 'cast' rule.
Arcsin, arccos, arctan
Arcsin is another way of writing the inverse of sin, arccos means the inverse of cos and arctan means the inverse of tan. For example, arcsin(0.5) = 30º . However, although this is true, we also know that sin(150º) = 0.5 (using the idea of related angles and the 'cast rule'). If we continue moving round the 'unit circle' (the circle with radius 1 that we have been drawing angles on above), then we find that sin(390º) is also 0.5 .
So we can write arcsin(0.5) = 30º, 150º, 390º, ...
Solving Equations
Example:
Solve the equation sinø = 0.6428, for 0 < ø < 360º
therefore ø = arcsin(0.6428)
= 40º, 140º, 400º, ...
but the question asks for solutions between 0 and 360º, so the answer is 40º and 140º .
Graphing sinø, cosø and tanø
The following are graphs of sinø, cosø and tanø:
Points to note:
The graphs of sinø and cosø are periodic, with period of 360º (in other words the graphs repeat themselves every 360º .
The graph of cosø is the same as the graph of sinø, although it is shifted 90º to the right/ left. For this reason sinø = cos(90 - ø) or cosø = sin(90 - ø).
The graph of tanø has asymptotes. An asymptote is a line which the graph gets very close to, but does not touch. The red lines are asymptotes.
These graphs obey the usual laws of graph transformations.
Sine and cosine formulae
Sine and Cosine Formulae
sin x = sin (180 - x)
cos x = -cos (180 - x)
The Sine Rule
This works in any triangle:
a = b = c
sinA sinB sinC
alternatively, sinA = sinB = sinC
a b c
NOTE: the triangle is labelled as follows:
The Cosine Rule
c² = a² + b² - 2abcosC
which can also be written as:
a² = b² + c² - 2bccosA
This also works in any triangle.
The area of a triangle
The area of any triangle is ½ absinC (using the above notation)
This formula is useful if you don't know the height of a triangle (since you need to know the height for ½ base × height).
