The Definite Integral

The Definite Integral

The Definite Integral

• The expression  $\int_a^b\ f\left(x\right)\ dx$  is the definite integral from a to b.

• a and b are called the limits of integration.

The Definite Integral

• Suppose that  $F\left(x\right)$  is the antiderivative of  $f\left(x\right)$ . That is ,

•  $\int_{ }\ f\left(x\right)\ dx\ =\ F\left(x\right)\ +\ C$ .

• Then  $\int_a^b\ f\left(x\right)\ dx\ =\ F\left(b\right)\ -\ F\left(a\right)$ .

Notes

• The order of the limits of integration matter. That is,

•  $\int_a^bf\left(x\right)\ dx\ \ne\int_b^a\ f\left(x\right)dx$ .

• There is no need to add C at the end, since it will get subtracted off anyway.

The Definite Integral

The next slide is an introduction video to the definite integral from The Organic Chemistry Tutor.

Theses will be your notes today.

Poll

How comfortable do you feel so far with the definite integral?

• This makes sense.

• I need more practice.

• I need additional help on this topic.

• I'm don't know what an integral is, and this point I'm too afraid to ask.

Multiple Choice

Practice 1: Evaluate $\int_1^4\left(3x^2\ -\ 4x\right)\ dx$  .

• 33

• 32

• -1

• 16

Multiple Choice

Practice 2: Evaluate  $\int_{-1}^3\ \left(12x\ +\ 4\right)\ dx$ .

• 64

• 66

• 32

• 6