Paul's Online Notes
Home / Calculus III / Surface Integrals / Surface Integrals
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 17.3 : Surface Integrals

1. Evaluate $$\displaystyle \iint\limits_{S}{{z + 3y - {x^2}\,dS}}$$ where $$S$$ is the portion of $$z = 2 - 3y + {x^2}$$ that lies over the triangle in the $$xy$$-plane with vertices $$\left( {0,0} \right)$$, $$\left( {2,0} \right)$$ and $$\left( {2, - 4} \right)$$. Solution
2. Evaluate $$\displaystyle \iint\limits_{S}{{40y\,dS}}$$ where $$S$$ is the portion of $$y = 3{x^2} + 3{z^2}$$ that lies behind $$y = 6$$. Solution
3. Evaluate $$\displaystyle \iint\limits_{S}{{2y\,dS}}$$ where $$S$$ is the portion of $${y^2} + {z^2} = 4$$ between $$x = 0$$ and $$x = 3 - z$$. Solution
4. Evaluate $$\displaystyle \iint\limits_{S}{{xz\,dS}}$$ where $$S$$ is the portion of the sphere of radius 3 with $$x \le 0$$, $$y \ge 0$$ and $$z \ge 0$$. Solution
5. Evaluate $$\displaystyle \iint\limits_{S}{{yz + 4xy\,dS}}$$ where $$S$$ is the surface of the solid bounded by $$4x + 2y + z = 8$$, $$z = 0$$, $$y = 0$$ and $$x = 0$$. Note that all four surfaces of this solid are included in $$S$$. Solution
6. Evaluate $$\displaystyle \iint\limits_{S}{{x - z\,dS}}$$ where $$S$$ is the surface of the solid bounded by $${x^2} + {y^2} = 4$$, $$z = x - 3$$, and $$z = x + 2$$. Note that all three surfaces of this solid are included in $$S$$. Solution