Integration

Integration

1. Integration is reverse of differentiation. Given that

y=mx +c then dy = m y=mx +c

dx

2. Integration divided into:

A) Indefinite Integral-no upper and lower limit must write constant C

B) Definite Integral-contains upper and lower limitno need to write constant C

3. How to integrate: same technique for definite and indefinite integral

a)

b)

c)

Whereby a and c are constants

Example for Formula in (a)

Example for formula in (b) and (c)

Integration of the expression:

Example: Integrate the following

1.

2.

4.

5.

6.

7.

8.

9.

Finding equation of curve from gradient function

Gradient function: i.e result of differentiation

For Example: Example:

Step 1: Integrating the gradient function

Step 2: Finding the constant C by substituting the given point or coordinate

Step 3: Rewrite the equation with the value of constant C

For example:

Step 1:

Step 2:

Step 3:

Solve the following questions:

1.

2.

3.

4.

Homework:

1.

6.

7.

8.

Integration by substitution

When to use this method usually when the power of the expression is more than 2 where the expression cant be expanded

For example if y = ( x +2)2 one can expand the power and integrate each term separately.

If in the case the power is more than 2, i.e y= ( 2x + 1)3 it is not practical to expand the term three times and then integrate each term separately.

By using the substitution one can solve the above expressions.

Solve the following using the above method:

1.

Substitution method

Formula method

2.

Substitution method

Formula method

Definite Integral

Example of definite Integral

Solve the following: 2. Miscellaneous exercise involving definite integral:

1.

b.

c.

d.

Finding Area as Summation of Integration

Exercise on area

Finding Volume as Summation of Integration

Exercise on Volume

Enrichment exercise

Sometime the question may ask to express y in term of x by giving the gradient function

Students still have to follow the above steps to express y in term of x.

Take it as coordinate or point

Gradient function