564 CHAPTER 28. THE INTEGRAL ON SURFACES

and the increment of surface area is then∣∣∣∣∣∣ −2sinφ sinθ

2sinφ cosθ

0

×

 2cosφ cosθ

2cosφ sinθ

−2sinφ

∣∣∣∣∣∣dθdφ

=

∣∣∣∣∣∣ −4sin2

φ cosθ

−4sin2φ sinθ

−4sinφ cosφ

∣∣∣∣∣∣dθdφ = 4sinφdθdφ

Therefore, since the hemisphere corresponds to θ ∈ [0,2π] and φ ∈ [0,π/2], the integral towork is ∫ 2π

0

∫π/2

0

[12(2sinφ cosθ)2 +

(12(2sinφ cosθ +2cosφ)

)·

(2sinφ sinθ)+12(2sinφ sinθ)2cosφ

]4sin(φ)dφdθ

Doing the integration, this reduces to 163 π .

The important thing to notice is that there is no new mathematics here. That which isnew is the significance of a flux integral which will be discussed more in the next chapter.In short, this integral often has the interpretation of a measure of how fast something iscrossing a surface.

28.6 Exercises1. Find a parametrization for the intersection of the planes 4x+ 2y+ 4z = 3 and 6x−

2y =−1.

2. Find a parametrization for the intersection of the plane 3x+y+z = 1 and the circularcylinder x2 + y2 = 1.

3. Find a parametrization for the intersection of the plane 3x + 2y+ 4z = 4 and theelliptic cylinder x2 +4z2 = 16.

4. Find a parametrization for the straight line joining (1,3,1) and (−2,5,3).

5. Find a parametrization for the intersection of the surfaces 4y + 3z = 3x2 + 2 and3y+2z =−x+3.

6. Find the area of S if S is the part of the circular cylinder x2 + y2 = 4 which liesbetween z = 0 and z = 2+ y.

7. Find the area of S if S is the part of the cone x2 +y2 = 16z2 between z = 0 and z = h.

8. Parametrizing the cylinder x2+y2 = a2 by x = acosv,y = asinv,z = u, show that thearea element is dA = adudv

9. Find the area enclosed by the limacon r = 2+ cosθ .

10. Find the surface area of the paraboloid z = h(1− x2 − y2

)between z = 0 and z = h.

Take a limit of this area as h decreases to 0.