Module MA2E02 (Frolov), Multivariable Calculus
Tutorial Sheet 7
Due: at the end of the tutorial session Tuesday/Thursday, 15/17 March 2016
Name and student number:
1. Consider the surface that extends upward from the semicircle y =
plane to the surface
1
y2
z = πx2 y +
12
3
(a) Make a rough sketch of the surface.
(b) Use a line integral to find the area of the surface.
Show the details of your work.
√
9 − x2 in the xy-
2.
(a) Let the path C between the points (−π/2 , π) and (3π/2 , −2π/3) be a curve formed from
two line segments C1 and C2 , where C1 is joining (−π/2 , π) and (3π/2 , π), and C2 is joining
(3π/2 , π) and (3π/2 , −2π/3).
i. Plot the path C, and show its orientation on the plot.
ii. Parameterise C1 , and evaluate
Z
3y 3
2
2
4
dy .
I1 ≡
6π sin(3x) − 8xy dx − 8x y + 3π cos
2
C1
iii. Parameterise C2 , and evaluate
Z
3y 3
2
2
4
I2 ≡
6π sin(3x) − 8xy dx − 8x y + 3π cos
dy .
2
C2
iv. Compute the sum I = I1 + I2 .
(b) Show that for any integration path C the integral above depends only on the initial and
terminal points (−π/2 , π) and (3π/2 , −2π/3) of the path C.
(c) Find the potential function φ(x, y).
(d) Use the Fundamental Theorem of Line Integrals to find the value of the integral above for an
integration path between the initial point (−π/2 , π) and the terminal point (3π/2 , −2π/3).
Show the details of your work.
3. Consider the integral
I
(3 sin x − 4xy + y 2 ) dx − (4e−y + x2 ) dy .
C
1
Assume that the curve C is oriented counterclockwise, and it is the boundary of the region R
between y = x2 /2 and y = x.
(a) Sketch the region R.
(b) Use Green’s Theorem to evaluate the integral.
Show the details of your work.
2