Unit V

Multiple Integral and special function

Q1) Evaluate the following:

A1)

=

=

=

Q2) Evaluate the following:

A2) let I = .dxdy

=

=

Q3) Evaluate the following triple integral

A3) let I = .dzdydx

=

=

=

=

Q4) Evaluate by changing to polar co-ordinates hence evaluate

A4) let I =

Polar form: x = rcos, y= rsin dx dy = r. Dr d

=

=

= ½

= -1/2

= - ½

= -1/2 = ½ (

= …(1)

To find

By equation 1.,

= /2.

Q5) Express in polar cartesian form hence evaluate it

A5) let cartesian form I= where

X varies from x = y to x = a

Y varies from y = 0 to y = a

Polar form: x = rcos, y= rsin dx dy = r. Dr d

varies from

I =

= r.dr. d

=

=

=

=a

= a

=

Q6) Plot the point with cylindrical co-ordinates (4, 2/3, -2) and express its location in rectangular coordinates.

A6) conversion from cylindrical to rectangular coordinates requires a simple application.

x = r cos = 4 cos (2/3) = -2

y = r sin = 4sin (2/3) = 2/.

Z = -2

The point with cylindrical co-ordinates (4, 2/3, -2) has rectangular coordinates (-2, 2/, -2) seen in the diagram.

Q7) Convert the rectangular co-ordinates (1, -3, 5) to cylindrical co-ordinates.

A7) use the second set of equations from note to translate from rectangular to cylindrical co-ordinates

r =

=

We choose the positive square root, so r = . Now, we apply the formula to find . In this case, y is negative and x is positive, which means we must select the value between and 2

Tan = =

= arctan (-3) 5.03 rad.

In this case the z-coordinates are the same in both rectangular and cylindrical coordinates: z = 5.

The point with rectangular co-ordinates (1, -3, 5) has cylindrical coordinates approximately equal to (

Q8) f(B) = Solve the given function.

A8) =

= [Recursive function for the gamma function]

= [Recursive formula for the gamma function]

=

= [By the definition of Beta function]

=

Q9) B =

A9) =

=

=

= [because ]

Q9) Evaluate I =

A10) let x2 = a2 y, we get

I =

= )

= .

Q10) Evaluate I =

A11) let

I =

=

= .

Q11) Prove that

A12) let I =

Put

=

= .

= = ¼. = .