Advanced Calculus: Problem sheet 6 Please hand in solutions by by 1pm on 21 May 2001.

When performing double integrals, it is normally helpful to sketch the region of integration. Also, it is sometimes easier to evaluate double integrals by first interchanging the order of integration.

1. Find the gradient of the following functions at the given point and calculate the directional derivative in the direction of the vector a.

(a) f(x, y) = y + x^{2}/y at (1,2), a = 12i + 5j.

(b) g(x, y) = x^{2} − 3xy + 2y^{2} at (1,1), a = 3i − 4j.

(c) h(x, y) = x tan^{−1}(y/x) at (1,−1), a = 2i + j.

2. Find and classify all the critical points of the function

f(x, y) = x^{3} + 3xy^{2} − 3x^{2} − 3y^{2} − 2.

3. Find and classify all the critical points of the function

g(x, y) = 15x − 9x^{2} − 2y^{2} + x^{3} + xy^{2}.

4. Find and classify all the critical points of the function

h(x, y) = x^{2} + y^{2} − 6x^{3}y^{3}.

5. Find the maximum and minimum values of the function F (x, y) = x^{4} + y^{4} on the

ellipse

x^{2} + xy + y^{2} = 3.

6. Find the maximum and minimum values of the function G(x, y, z) = xy^{2}z^{3} on the

sphere

x^{2} + y^{2} + z^{2} = 6.

7. Find the point on the paraboloid z = x^{2} + y^{2} which is closest to the point (3, 3, 1). [Hint: The distance from (x, y, z) to (3, 3, 1) is \sqrt{(x − 3)^{2} + (y − 3)^{2} + (z − 1)^{2}}. Minimize the square of this distance for (x, y, z) on the paraboloid.]

8. Evaluate ∫_{1}^{ln 2}∫_{0}^{ln y} e^{x−y} dx dy.

9. Evaluate (a) ∫_{0}^{3}∫_{0}^{2}(2x − y)^{2} dy dx, (b) ∫_{0}^{π/2}∫_{sin θ}^{1} r sin θ dr dθ.

10. Evaluate ∫_{0}^{π}∫_{0}^{π} | sin(x + y)| dx dy.

11. By changing the order of integration or otherwise, evaluate ∫_{0}^{a} ∫_{x}^{a} \frac{x}{\sqrt{(x^{2} + y^{2})}} dy dx

where a > 0.

12. Us polar coordinates to evaluate ∫_{−2}^{2} ∫_{0}^{√\overline{(4−y^{2})}}(x^{2} + y^{2})^{2} dx dy.

13. Evaluate the integral ∫ ∫_{T} (x^{2} − y^{2}) dA over the triangle T with vertices (0, 0), (1, 0) and (1, 1).

14. Evaluate the integral ∫ ∫_{T} (x + y)^{3} dA over the triangle T with vertices (0, 1), (1, 0) and (2, 1).

15. Evaluate the integral ∫ ∫_{R} \frac{xy}{(x^{2} + y^{2})^{2}} dA over the region R in the first quadrant

bounded by the circles x^{2} + y^{2} = 1, x^{2} + y^{2} = 4 and the lines x = 0 and y = x.

16. Find the volume of the solid that lies below the surface z = x^{2} + y^{2} and above the bounded region in the (x, y)-plane enclosed by the lines y = 0 and the curve y = 1 − x^{2}.

17. Evaluate the integral ∫ ∫ ∫_{R} x^{3}y^{2}z dV where R is the finite region bounded by the surface z = xy and the planes y = x, x = 1 and z = 0.

18. Find the volume of the solid that lies below the plane z = x + 4 and above the bounded region in the (x, y)-plane enclosed by the parabola y = 3 − x^{2} and the line y = 2x.

19. Use the transformation u = x − y, v = x + y to evaluate

∫_{0}^{1}∫_{0}^{1−y} exp[(x − y)/(x + y)] dx dy.

20. Evaluate

∫ ∫_{R} xy\sqrt{(1 − x^{4} − y^{4})} dx dy,

where R is the region x ≥ 0, y ≥ 0, x^{4} + y^{4} ≤ 1, by using the transformation x^{2} = u cos v, y^{2} = u sin v.

RJC 30/4/2001