### Finding the Constant of Integration

**What is the constant of integration?**

- When finding an
**anti**-**derivative**there is a constant term to consider- this constant term, usually called , is the
**constant**of**integration**

- this constant term, usually called , is the
- In terms of
**graphing**an**anti**-**derivative**, there are endless possibilities- collectively these may be referred to as the
**family**of**antiderivatives**or**family**of**curves** - the constant of integration is determined by the
**exact**location of the curve- if a
**point**on the**curve**is**known,**the**constant**of**integration**can be found

- if a

- collectively these may be referred to as the

**How do I find the constant of integration?**

- For , the
**constant**of**integration, -**and so the particular**antiderivative**- can be found if a point the graph of passes through is known

**STEP 1**

If need be, rewrite into an integrable form

Each term needs to be a power of (or a constant)

**STEP 2**

Integrate each term of, remembering the constant of integration, “”

(Increase power by 1 and divide by new power)

**STEP 3**

Substitute the and coordinates of a given point in to to form an equation in

Solve the equation to find

#### Exam Tip

- If a constant of integration can be found then the question will need to give you some extra information
- If this is given then make sure you use it to find the value of c

#### Worked Example

The graph of passes through the point . The gradient function of is given by.

Find.

### Area Under a Curve Basics

**What is meant by the area under a curve?**

- The phrase “
**area under a curve”**refers to the area bounded by- the graph of
- the -axis
- the
**vertical**line - the
**vertical**line

- The
**exact area****under****a****curve**is found by evaluating a**definite integral** - The graph of could be a
**straight****line**- the use of
**integration**described below would still apply- but the shape created would be a
**trapezoid** - so it is easier to use “”

- but the shape created would be a

- the use of

**What is a definite integral?**

- This is known as the
**Fundamental****Theorem****of****Calculus** **a**and**b**are called limits**a**is the**lower**limit**b**is the**upper**limit

- is the
**integrand** - is an
**antiderivative**of - The
**constant**of**integration**(“”) is not needed in**definite****integration**- "” would appear alongside both
**F(a)**and**F(b)** - subtracting means the “”’s cancel

- "” would appear alongside both

**How do I form a definite integral to find the area under a curve?**

- The graph of and the -axis should be obvious boundaries for the area so the key here is in finding and - the
**lower**and**upper**limits of the**integral**

**STEP 1**

Use the given sketch to help locate the limits

You may prefer to plot the graph on your GDC and find the limits from there

**STEP 2**

Look carefully where the ‘left’ and ‘right’ boundaries of the area lie

If the boundaries are vertical lines, the limits will come directly from their equations

Look out for the -axis being one of the (vertical) boudnaries – in this case the limit () will be 0

One, or both, of the limits, could be a root of the equation

i.e. where the graph of crosses the -axis

In this case solve the equation to find the limit(s)

A GDC will solve this equation, either from the graphing screen or the equation solver

**STEP 3**

The definite integral for finding the area can now be set up in the form

#### Exam Tip

- Look out for questions that ask you to find an
**indefinite**integral in one part (so “**+c**” needed), then in a later part use the same integral as a**definite**integral (where “**+c**” is not needed) - Add information to any diagram provided in the question, as well as axes intercepts and values of limits
- Mark and shade the area you’re trying to find, and if no diagram is provided,
**sketch**one!

- Mark and shade the area you’re trying to find, and if no diagram is provided,

### Definite Integrals using GDC

**Does my calculator/GDC do definite integrals?**

- Modern graphic calculators (and some ‘advanced’ scientific calculators) have the functionality to evaluate
**definite****integrals**- i.e. they can calculate the
**area under a curve**(see above)

- i.e. they can calculate the
- If a calculator has a button for evalutaing definite integrals it will look something like

- This may be a physical button or accessed via an on-screen menu
- Some GDCs may have the ability to find the area under a curve from the graphing screen
- Be careful with
**any**calculator/GDC, they may not produce an**exact**answer

**How do I use my GDC to find definite integrals?**

**Without graphing first …**

- Once you know the
**definite****integral**function your calculator will need three things in order to evaluate it- The function to be integrated (
**integrand**) () - The
**lower**limit ( from ) - The
**upper**limit ( from )

- The function to be integrated (
- Have a play with the order in which your calculator expects these to be entered – some do not always work left to right as it appears on screen!

**With graphing first ...**

- Plot the graph of
- You may also wish to plot the vertical lines and
- make sure your GDC is expecting an "" style equation

- Once you have plotted the graph you need to look for an option regarding “area” or a physical button
- it may appear as the integral symbol (e.g. )
- your GDC may allow you to select the lower and upper limits by moving a cursor along the curve - however this may not be very accurate
- your GDC may allow you to type the exact limits required from the keypad
- the lower limit would be typed in first
- read any information that appears on screen carefully to make sure

- You may also wish to plot the vertical lines and

#### Exam Tip

- When revising for your exams always use your GDC to check any definite integrals you have carried out by hand
- This will ensure you are confident using the calculator you plan to take into the exam and should also get you into the habit of using you GDC to check your work, something you should do if possible

#### Worked Example

a)

Using your GDC to help, or otherwise, sketch the graphs of

,

and

on the same diagram

b)

The area enclosed by the three graphs from part (a) and the -axis is to be found.

Write down an integral that would find this area.

c)

Using your GDC, or otherwise, find the exact area described in part (b).

Give your answer in the form where and are integers.