Integrals

Question 0

2. Indefinite Integrals

Answer 0

Indefinite integrals are used to find the primitive function a particular derivative came from.
They are expressed in the form:
The integral of f(x) dx

Question 1

1. Definite Integrals

Answer 1

Definite integrals are used to find the area between a function and a given domain.
They are expressed in the form:
The integral of f(x) dx from a to b

Question 2

Definite Integral Rule

Answer 2

∫x^n dx from a to b
= ∫x^n+1/n+1 from a to b
= b^n+1/n+1 - a^n+1/n+1

Question 3

Solving Indefinite Integrals

Answer 3

The integral of x^n dx= x^n+1/n+1 +C

Where C is a constant.

Question 4

The Reverse Chain Rule

Answer 4

The ∫f(x)^n
=1/f’(x) ∫f’(x) f(x)^n dx
=1/f’(x) (f(x)^n+1/n+1) + C

Question 5

Areas enclosed by the X-axis

Answer 5

Use definite integrals. (Assuming y is the subject)
IMPORTANT!!!!
Take the absolute values of areas in the negative x-axis zone.

Question 6

Areas enclosed by the Y-Axis

Answer 6

*Make X the subject
Use definite integrals for y:
∫f(y) dy from a to b
IMPORTANT!!!
Take the absolute value of the negative y-axis.

Question 7

Areas enclosed by two curves

Answer 7

If we are to find the area enclosed by two curves we must:

1. Sketch both curves
2. Shade the required area
3. Set Up a simultaneous equation to find any POIs
4. Use integration to find the required area i.e. Integration of top curve- integration of bottom curve from first POI to second POI

Question 8

Method 2 Lolliwrapper Method

Answer 8

*Rule
A= ∫f(x)-g(x) dx from a to b
Where f(x) is the top function and g(x) the bottom

Question 9

Volumes rotated about the x-axis

Answer 9

Vx-axis=π ∫f(x)^2 dx from a to b

Question 10

Volume about the y-axis

Answer 10

Vy-axis=π ∫f(y)^2 dy from a to b

—> Make y the subject

Question 11

Volumes between two lines

Answer 11

V= ∫f(x)^2-g(x)^2 dx from a to b 
Where f(x) is the top function and g(x) the bottom