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Chapter 13: Problem 45
Identify the quadric surface. $$ x^{2}-y+z^{2}=0 $$
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Evaluate the double integral. $$ \int_{0}^{2} \int_{3 y^{2}-6 y}^{2 y-y^{2}} 3 y d x d y $$
Exercises 55 and 56, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \int_{-1}^{1} \int_{-2}^{2} y d y d x=\int_{-1}^{1} \int_{-2}^{2} y d x d y $$
Sketch the region \(R\) whose area is given by the double integral. Then change the order of integration and show that both orders yield the same area. $$ \int_{0}^{1} \int_{y^{2}}^{\sqrt[3]{y}} d x d y $$
Set up the integral for both orders of integration and use the more convenient order to evaluate the integral over the region \(R\). $$ \begin{aligned} &\int_{R} \int x y d A\\\ &R \text { : rectangle with vertices at }(0,0),(0,5),(3,5),(3,0) \end{aligned} $$
Evaluate the partial integral. $$ \int_{0}^{x} y e^{x y} d y $$
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