Advance Calculus 2

Please use radian mode and π = 3.142.

Question 1

A circular cone over the region E is defined by,

Z=a and z =

The cone has height a, circular end of radius a, and the apex of the cone is at the origin. The projection of E onto the xy-plane is a circle x2+ y2 < a2 as shown in the diagram below. Evaluate the triple integral,

when a = 1.78, by first converting it to cylindrical polar coordinates. Give your answer to 3 decimal places. Take π = 3.142.

Answer:

Question 2

A 3-dimensional vector field F is given by, F(x,y,z) = z cosx i + siny j + y tan-1z k Calculate the divergence of F at the point (1.7,1.3,0.4) giving your answer to 3 decimal places.

Answer:

 

Question 3

Determine the scalar line integral,

- dr

where F = x2i + xy j. C is a path specified by the equation y = 1 - x2 from (0,0.64) to (0.64,0). Give your answer to 3 decimal places.

Answer:

Question 4

A 3-dimensional vector field F is given by, F = xyz3 i + 2xyz4 j - 2x2yz k

Calculate the j component of curl F at the point (2.1,2.2,0.6). Give your answer to 3 decimal places.

Answer:

Question 5  

A 3-dimensional scalar field is given by, f(x,y,z) = x2y - y2z + z2x At the point (8.8,6,6.9), find the derivative of f in the direction 4i + 5j + 3k. Give your answer to 3 decimal places.

Answer:

 

Question 6

A flat circular disc, of radius R, can be modeled as a thin disc of negligible thickness. It has a surface mass density function given by f(r,φ) = k(1 - r2/R2), where k is the surface density at the centre and r is the distance from the centre of the disc. Using area integral in plane polar coordinates, calculate the total mass of the disc, in kg, when R = 0.18 m and k = 23.22 kg m-2. Give your answer to 3 decimal places. Take π = 3.142.

Answer:

Question 7

A 3-dimensional scalar field is given by,

Calculate the k component of grad φ at the point (8.83,8.83,4.61). Give your answer to 3 decimal places.

Answer:

Question 8

Determine which one of the following vector fields is conservative.

		U = (y2 + 2z)i + (5xy + 6z)j + (2xz + y + z2)k
		H = -k(2x i + y j)     where k is a constant.
		G = 2x i - (2 - xz)j + xy k
		F = -w(y i - x j)    where w is a positive constant.

Question 9

The surface integral is given by,

where S is the region of the xy-plane given by

x ≥ 0, y ≥ 0, x + y ≤ t

where t is a constant. Evaluate this surface integral when t = 2.17, giving your answer to 3 decimal places.

Answer:

Question 10

A 3-dimensional scalar field g is given by, g(x,y,z) = x2y2z2 Find the maximum value of the derivative of g at the point (1.1,3.6,0.5). Give your answer to 3 decimal places.

Answer: