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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
to complete the evaluation.
Where possible hand written is welcome