Use parameterization to compute the area of paraboloid z = x^2 + y^2 that lies between plane z =...
Question:
Use parameterization to compute the area of paraboloid {eq}z = x^2 + y^2 {/eq} that lies between plane {eq}z = 1 {/eq} and {eq}z = 4 {/eq}.
Surface Integral:
When working with surfaces in parametrics it is necessary to calculate the surface differential for the calculation of surface integrals.
This differential is calculated through the cross product through the expression:
{eq}S \equiv \left\{ {\begin{array}{*{20}{c}} {x = x\left( {u,v} \right)} \\ {y = y\left( {u,v} \right)} \\ {z = z\left( {u,v} \right)} \end{array},\;} \right.\left( {u,v} \right) \in D \to dS = \left\| {\frac{{\partial S}}{{\partial u}} \times \frac{{\partial S}}{{\partial v}}} \right\|dudv {/eq}
Answer and Explanation: 1
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View this answerFirst, we parameterize the surface:
{eq}z = {x^2} + {y^2} \to S \equiv \left\{ {\begin{array}{*{20}{c}} {x = u\cos v}\\ {y = u\sin v}\\ {z =...
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