1 Assignment 2 [20]

Instructions:

Please answer the questions carefully, and clearly write your student number and name.

When uploading your assignment, upload the document as a pdf.

When saving your document, please save the file with your students number then name (such as ”201912345-JohnPaul.pdf”).

Note that the deadline to submit the assignment is 08:00 on Friday 29 May 2020.

Question 1 [4]

Use Lagrange multipliers to find the point (a,b) on the graph of y = e6x, where the value of ab is as small as possible.

Question 2 [4]

Let f(x,y) = x2ex2 and let R be the triangle bounded by the lines x = 4, x = y/2, and y = x in the xy-plane. Express RR f dA as a double integral in two different ways (dxdy and dy dx), then evaluate one of your integrals to find the value of RR f dA.

Question 3 [2]

Convert the integral

to polar coordinates.

Question 4 [4]

Find the volume of the solid in R3 bounded by y = x2, x = y2, z = x + y + 24 and z = 0.

(Show calculations)

Question 5 [3]

Set up the integral RRRW f(x,y,z)dV for the function f(x,y,z) = z and region x2 + y2 = z = 49 in cylindrical coordinates.

Question 6 [3]

Convert the integral

to spherical coordinates.

1

Instructions:

Please answer the questions carefully, and clearly write your student number and name.

When uploading your assignment, upload the document as a pdf.

When saving your document, please save the file with your students number then name (such as ”201912345-JohnPaul.pdf”).

Note that the deadline to submit the assignment is 08:00 on Friday 29 May 2020.

Question 1 [4]

Use Lagrange multipliers to find the point (a,b) on the graph of y = e6x, where the value of ab is as small as possible.

Question 2 [4]

Let f(x,y) = x2ex2 and let R be the triangle bounded by the lines x = 4, x = y/2, and y = x in the xy-plane. Express RR f dA as a double integral in two different ways (dxdy and dy dx), then evaluate one of your integrals to find the value of RR f dA.

Question 3 [2]

Convert the integral

to polar coordinates.

Question 4 [4]

Find the volume of the solid in R3 bounded by y = x2, x = y2, z = x + y + 24 and z = 0.

(Show calculations)

Question 5 [3]

Set up the integral RRRW f(x,y,z)dV for the function f(x,y,z) = z and region x2 + y2 = z = 49 in cylindrical coordinates.

Question 6 [3]

Convert the integral

to spherical coordinates.

1

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