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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 17.4 : Surface Integrals of Vector Fields
- Evaluate \( \displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}\) where \(\vec F = \left( {z - y} \right)\,\vec i + x\,\vec j + 4y\,\vec k\) and \(S\) is the portion of \(x + y + z = 2\) that is in the 1st octant oriented in the positive \(z\)-axis direction.
- Evaluate \( \displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}\) where \(\vec F = \left( {x - 4} \right)\,\vec i + z\,\vec j - y\,\vec k\) and \(S\) is the portion of \(x = 4 - {y^2} - {z^2}\) that lies in front of \(x = - 2\) oriented in the negative \(x\)-axis direction.
- Evaluate \( \displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}\) where \(\vec F = \,\vec i + 4z\,\vec j + \left( {z - y} \right)\,\vec k\) and \(S\) is the portion of \(y = 4z + {x^3} + 6\) that lies over the region in the xz-plane with bounded by \(z = {x^3}\), \(x = 1\) and the x-axis oriented in the positive \(y\)-axis direction.
- Evaluate \( \displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}\) where \(\vec F = \left( {x + y} \right)\,\vec i + x\,\vec j + z{x^2}\,\vec k\) and \(S\) is the portion of \({x^2} + {y^2} = 36\) between \(z = - 3\) and \(z = 1\) oriented outward (i.e. away from the \(z\)-axis).
- Evaluate \( \displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}\) where \(\vec F = z\,\vec i + 3\,\vec k\) and \(S\) is the portion of \({x^2} + {y^2} + {z^2} = 4\) with \(z \ge 0\) oriented outwards (i.e. away from the origin).
- Evaluate \( \displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}\) where \(\vec F = - x\,\vec i + \left( {4 + y} \right)\,\vec j - z\,\vec k\) and \(S\) is the portion of \({x^2} + {z^2} = 9\) between \(y = 2\) and \(y = 10 - x\) oriented inward (i.e. towards from the \(y\)-axis).
- Evaluate \( \displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}\) where \(\vec F = y\,\vec i + 2\,\vec j + {\left( {z + 3} \right)^2}\,\vec k\) and \(S\) is the surface of the solid bounded by \(z = 2{x^2} + 2{y^2} - 3\) and \(z = 1\) with the negative orientation. Note that both surfaces of this solid are included in \(S\).
- Evaluate \( \displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}\) where \(\vec F = \left( {x - y} \right)\,\vec i + z\,\vec j + y\,\vec k\) and \(S\) is the surface of the solid bounded by \({y^2} + {z^2} = 4\), \(x = y - 3\), and \(x = 6 - z\) with the positive orientation. Note that all three surfaces of this solid are included in \(S\).
- Evaluate \( \displaystyle \iint\limits_{S}{{\vec F\centerdot \,d\vec S}}\) where \(\vec F = y\,\vec i - 2\,\vec k\) and \(S\) is the portion of the sphere of radius 1 with \(z \ge 0\) and \(x \le 0\) with the positive orientation. Note that all three surfaces of this solid are included in \(S\).