MAST20029 Engineering Mathematics Semester 2 2022 Assignment 1
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MAST20029 Engineering Mathematics
Semester 2 2022
Assignment 1
1. Consider the double integral
2≠ 1
I = j (t.╱) d╱ dt
+ cos Y
(a) Sketch the region of integration.
(b) Calculate the area of the region of integration.
(c) Obtain a double integral equivalent to I but with the order of integration reversed.
2. Let V be the solid region in R3 bounded above by the cone 之 = ·^t2 + ╱ 2 and bounded below by the sphere t2 + ╱ 2 + 之2 = 9.
(a) Sketch the region V .
(b) Calculate the volume of V by using spherical coordinates.
(c) Calculate the volume of V by using cylindrical coordinates.
(d) Calculate the surface area of the part of V that lies on the sphere t2 + ╱ 2 + 之2 = 9 and for which 之 > ·5/2, by solving an appropriate double integral.
(e) Verify your answer to part (d) by computing the double integral using MATLAB.
3. Let s be the part of the surface t2 +╱2 + 之2 = 1 such that ╱ < 0 and 之 > 0, oriented with upwards unit normal. Let C be the boundary of s, oriented anti-clockwise when viewed from above.
(a) Sketch the surface s, indicating the orientation of s and C.
(b) Give a parametrisation for C in terms of a parameter f, with f increasing, beginning and ending at the point (1.0.0).
(c) Using your parametrisation in part (b), use MATLAB to plot the curve C.
(d) Find the flux of the vector field
F(t.╱.之) = 之j + e^Y2 ′u2 k
across s.
2022-08-18