Section 7.7: Improper Integrals

If makes sense, then what about ?1

3

2

1dx

x 21

1dx

x

Definition1 1

( ) ( )limN

Ndx dxf x f x

This is our first example of an improper integral.

21

1dx

x

1

1dxx

1 1lim

1N N

2

1

1lim

N

N

dxx

|lim ln | |1|lnN

N

1

1

1lim

N

N

dxx

If the limit is finite then we say thatthe improper integral converges

otherwise the integral diverges.

Notice that if the integral is going to converge, Zero must be a horizontal asymptote

and the function must get small quickly.

Fact1

1 converges 1

pdx

xp

1

1dxx

lim 2 1 2N

N

Vertical Asymptotes provide another kind of improper integral.You always have to check that the function is continuous.

3

0

1

1dx

x

Diverges, but if we ignore the asymptote, we get the wrong answer:

30ln | 1| ln 2x

1lim ln | 1| ln | 0 1|N

N

3

01 1

lim lim1 1

N

N NN

dx dx

x x

1lim ln | 1| ln | 3 1|N

N

Third Type

2

1

1dx

x

0

2 20

1 1

1 1dx dx

x x

02

1 1tan (0)lim ( )tanN

N

1 1tan ( )lim (0tan )

NN

02

Volume =2

1

1dx

x

This is a paint can that can be filled with π unit3 of paint, but the surface requires an infinite amount of paint!

Gabriel’s hornRotate f(x) = 1/x about x-axis

Surface Area =2

21

1 12 1 dx

x x

Diverges!!