(a) from (0,0) to (1,2) along

Some integrations look simple but turn out to be quite involved. Question 1 covers such a situation. In such questions it can help to split the working into two vertical columns, with the main integration being dealt with in the LH column and the substitution working being dealt with in the RH column, in parallel.

1.    Find the length of the curve   between   and  . To have confidence in your answer calculate the actual distance between the points (1,1) and (5,25) and compare the two values.
(Hint: use     to get around the square root problem, in the first instance, and then use the hyperbolic identity (see attached sheet)     to complete the integration. Remember to change the limits of integration from the x values to the corresponding θ values.)

2.    Find the mean value of   between   and  .
(Hint: use the trigonometric identity     to “ease” the integration.)

3.    Find and classify the stationary points of
  .
Classify them as maximum, minimum or saddle points.

4.    Find the extrema of the surface   subject to the constraint  .

5.    Find the volume of the ellipsoid described by  .
(Hint: take advantage of the y/z symmetry and consider the elementary xy plane (so z=0) circular disc cross-section, of thickness δx for an arbitrary value of x, and sum all such sections.)

6.    Show that   is an exact differential. Hence find the potential function   which gives rise to this exact differential.

7.    Use the directional derivative to find the slope of the surface     at P(1,1) in the direction θ=35° (measured anticlockwise from the positive x axis). Compare your answer to the slope of the surface in the positive x, and positive y directions.

8.    Evaluate the integral     along the two separate paths:
(a)    from (0,0) to (1,2) along   
(b)    from (0,0) to (1,0) along   , and then from (1,0) to (1,2) along   .

9.    Sketch the domain of integration D corresponding to the repeated integral
 

Change the order of the integration and hence evaluate   .

10.    Evaluate     where the domain of integration D is in the first quadrant (x≥0, y≥0) and is bounded by
 ,   ,     and    .
Sketch the domain of integration and use the change of variables
   and   
to complete the evaluation.

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