Problems

1            Find the area of the triangle cut from the plane x + 2y + 4z = 10 by the coordinate planes.

2            Find the area cut from the plane 2x + 4y + z = 0 by the cylinder x² + y² = 1.

3            Find the area of the surface of the paraboloid z = x² + y² below the plane z = 1.

4            Find the area of the surface of the cone below the plane z = 2.

5            Find the surface area of the part of the sphere x² + y² + z² = a² which lies in the first octant; i.e., x ≥ 0, y ≥ 0, z ≥ 0.

6            Find the surface area of the part of the sphere x² + y² + z² = a² which is above the circle x² + y² ≤ b² (b ≤ a).

7            Find the surface area cut from the hyperboloid z = x² - y² by the cylinder x² + y² = a².

8            Find the area cut from the surface z = xyby the cylinder x² + y² = a².

9            Find the surface area of the part of the sphere r² + z² = a² above the circle r = a cos θ.

10            Find the surface area of the part of the cone z = cr above the circle r = a cos θ.

11            Find the area of the part of the plane z = ax + by + c over a region D of area A.

12            Find the surface area of the part of the cone over a region D of area A.

13            Find the surface area of the part of the cylinder x² + z² = a² cut out by the cylinder

x² + y² ≤ a².

14            Find the surface area of the part of the cylinder x² + z² = a² above and below the square -b ≤ x ≤ b, -b ≤ y ≤ b (b ≤ a).

15            Evaluate the surface integral

where S is the surface z = x² + y², -1 ≤ x ≤ 1, - 1 ≤ y ≤ 1, oriented with the top side positive.

16            Evaluate the surface integral

where S is the surface z = 3x - 5y over the rectangle 1 ≤ x ≤ 2, 0 ≤ y ≤ 2, oriented with the top side positive.

17            Evaluate the surface integral

where S is the surface z = 1 - x² - y², x² + y² ≤ 1, oriented with the top side positive.

18            Evaluate the surface integral

where S is the surface z = x + y² + 2, 0 ≤ x ≤ 1, x ≤ y ≤ 1, oriented with the top side positive.

19            Evaluate the surface integral

where S is the surface z = xy, 0 ≤ x ≤ 1, - ,x ≤ y ≤ x, oriented with the top side positive.

20            Evaluate the surface integral

where S is the surface , x² + y² ≤ b, oriented with the top side

positive (b < a).

21             Show that if S is a horizontal surface z = c over a region D, oriented with the top side positive, then the surface integral over S is