Properties of Integrals

Suppose and are integrable on .

Flipping the range

Addition

will be integrable on .

Converse is not true.

Proof Hint

• Prove is integrable using:
• Start with and show
• Start with and show

Constant multiplication

Suppose . will be integrable .

Converse is not true.

Proof Hint

• Prove for . Use
• Prove for
• Using the above results, proof for is apparent

Bounds

If on :

If on :

Modulus

will be integrable on .

Proof Hint

Start with . And integrate both sides.

Multiple

will be integrable on . Converse is not true.

Proof Hint

• Suppose is bounded by
• Prove is integrable (Use )
• is integrable because:

Max, Min

and are integrable.

Where and functions are defined as:

Additivity

is Riemann integrable on where .

Proof Hint

• : Use Cauchy criterion after defining these:
• : Use cauchy criterion on separately and then combine using a union partition

After the integrability is proven,

Proof Hint

1. Let be a partition on and be a partition on . And .

2. Prove the below using Cauchy criteria:

3. Prove the below using Cauchy criteria (by considering RHS):