Applications of Integration

Applications of Integration

In my previous posts, we discussed Definite and Indefinite Integrations. Now we shall learn about Applications of Derivatives. Initially, we shall discuss “Area Under Curves”.

Area Under Curve-: If we want to calculate the area between the curves y=f(x) and y=g(x) then there are actually two cases-

First Case when   f(x) \ge g(x)Below is the figure showing this case

Area_G1

here area under these  two curves       

 

The second Case When  f(x) \le g(x)Below figure shows this case

Area_G2

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Indefinite Integration-Topics in IB Mathematics

Indefinite Integration

After a long series on differentiation and ‘Application of derivatives‘, we shall now discuss Indefinite Integration. It consists of two different words indefinite and integration.
First of all, we shall learn about Integration.

 Integration is the reverse process of differentiation so we can also call it as antiderivative. There is one more name for it, that is Primitive.
If f & g are functions of x such that g'(x) = f(x) then the function g is called a Primitive Or Antiderivative Or Integral of  f(x) w.r.t. x and is written symbolically as:-

\int {f\left( x \right){\rm{ }}dx = g\left( x \right) + c}

If    \frac{d}{{dx}}\left\{ {f(x) + c} \right\} = f'(x)

then  \int {f'\left( x \right){\rm{ }}dx = f\left( x \right) + c}      here c is just an arbitrary constant. Value of c is not definite that’s why we call it Indefinite Integration.

Techniques  Of  Integration-: There are a few important techniques used to solve problems based on integration

(i) Substitution or  Change of Independent Variable- If the derivative of a function is given in the question, then we should use the method of substitution to integrate that question. Read more

Applications of Derivatives in IB Mathematics

Applications of Derivatives in IB

Mathematics-

In my previous post, we discussed how to find the derivative of different types of functions as well as the geometrical meaning of differentiation. Here we are discussing  Applications of Derivatives in IB Mathematics
There are many different fields for the Applications of Derivatives. We shall discuss a few of them-

Slope and Equation of tangents to a curve- If We draw a tangent to a curve y=f(x) at a given point   ({x_1},{y_1}), then

The gradient of the curve at given point=the gradient of the tangent line  at given  point

and we already discussed that slope or gradient of the tangent at given point   ({x_1},{y_1})

m=  {\frac{{dy}}{{dx}}_{at({x_1},{y_1})}}

=f'({x_1})

Finally to find the equation of tangent we use the slope-point form of equation

y - {y_1} = m(x - {x_1})

The major part of this concept is also discussed in the previous post. We should also remember following points while solving these types of questions.

(i) If two lines are parallel to each other, their slopes are always equal
i.e     {m_1} = {m_2}
(ii) If two lines are perpendicular to each other, the product of their  slopes is always -1

{m_1}.{m_2} = - 1

(iii) If a line is passing through two points   ({x_1},{y_1}) and  ({x_2},{y_2})  then, slope of the line

m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}

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