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Chapter 13: Q. 34 (page 1015)

In Exercises 29鈥�34, sketch the region determined by the limits of the iterated integrals and then give another iterated integral (or a sum of iterated integrals if necessary) using the opposite order of integration
${鈭�}_{0}^{8} {鈭�}_{\sqrt[3]{y}}^{2} f (x, y) d x d y$

Short Answer

Expert verified

The sketch of the region:

The order of integral is changed as:

${鈭�}_{0}^{2} {鈭�}_{0}^{x^{3}} f (x, y) d y d x$

Step by step solution

01

Step 1. Given information

Integral:

${鈭�}_{0}^{8} {鈭�}_{\sqrt[3]{y}}^{2} f (x, y) d x d y$

02

Step 2. Sketch the region

In the given region we can see that

$\sqrt[3]{y} 鈮� x 鈮� 2 0 鈮� y 鈮� 8$

The sketch of the region is:

03

Step 3. Change order of integral

In the above graph we can see that $0 鈮� y 鈮� x^{3}$ then $0 鈮� x 鈮� 2$

so the order of integral is changed as:

${鈭�}_{0}^{2} {鈭�}_{0}^{x^{3}} f (x, y) d y d x$

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Most popular questions from this chapter

$Express the sum \frac{2}{5} + \frac{2}{6} + \frac{2}{7} + \frac{2}{8} + \frac{4}{5} + \frac{4}{6} + \frac{4}{7} + \frac{4}{8} using double - summation notation .$

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.

${鈭�}_{0}^{2} {鈭�}_{0}^{3} {鈭�}_{4 x / 3}^{4} f (x, y, z) d z d x d y$

Evaluate each of the integrals in exercise 33-36 as iterated integrals and then compare your answers with those you found in exercise 29-32

$鈭� {鈭�}_{R} x^{2} y^{3} d A W h e r e R = {(x, y) | 1 鈮� x 鈮� 3, 0 鈮� y 鈮� 2}$

$L e t R = \{(x, y, z) | a_{1} 猢� x 猢� a_{2}, b_{1} 猢� y 猢� b_{2}, c_{1} 猢� z 猢� c_{2}\} . P r o v e t h a t : {鈭�}_{R} d V = (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) . W h a t i s t h e r e l a t i o n s h i p b e t w e e n R a n d t h e p r o d u c t (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) .$

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane

$f (x, y) = - 2 x^{2} y^{3} W h e r e R = {(x, y) | - 2 鈮� x 鈮� 3, - 1 鈮� y 鈮� 5}$

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