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Chapter 13: Q. 50 (page 1016)

In Exercises 49鈥�58, sketch the region determined by the iterated integral and then evaluate the integral. For some of these integrals, it may be helpful to reverse the order of integration.
${鈭�}_{0}^{3} {鈭�}_{y + 2}^{2 y - 3} (x^{2} + 3 x y) d x d y$

Short Answer

Expert verified

Value of the given integral is: $- \frac{1135}{8}$

Step by step solution

01

Step 1. Given Information

An integral, ${鈭�}_{0}^{3} {鈭�}_{y + 2}^{2 y - 3} (x^{2} + 3 x y) d x d y$

02

Step 2. Sketching the region of integration

Region of integration is shaded in the graph below.

03

Step 3. Evaluating the given iterated integral

Given integral,

{鈭�}_{0}^{3} {鈭�}_{y + 2}^{2 y - 3} (x^{2} + 3 x y) d x d y = {鈭�}_{0}^{3} {(\frac{x^{3}}{3} + \frac{3 x^{2} y}{2})}_{y + 2}^{2 y - 3} d y = {鈭�}_{0}^{3} (\frac{{(2 y - 3)}^{3} - {(y + 2)}^{3}}{3} + \frac{3 y [{(2 y - 3)}^{2} - {(y + 2)}^{2}]}{2}) d y = {鈭�}_{0}^{3} (\frac{(8 y^{3} - 27 - 36 y^{2} + 54 y) - (y^{3} + 8 + 6 y^{2} + 12 y)}{3} + \frac{3 y [(4 y^{2} + 9 - 12 y) - (y^{2} + 4 + 4 y)]}{2}) d y = {鈭�}_{0}^{3} (\frac{(7 y^{3} - 35 - 42 y^{2} + 42 y)}{3} + \frac{3 y (3 y^{2} + 5 - 16 y)}{2}) d y = {鈭�}_{0}^{3} (\frac{7 y^{3} - 42 y^{2} + 42 y - 35}{3} + \frac{9 y^{3} - 48 y^{2} + 15 y}{2}) d y = {(\frac{\frac{7 y^{4}}{4} - \frac{42 y^{3}}{3} + \frac{42 y^{2}}{2} - 35 y}{3} + \frac{\frac{9 y^{4}}{4} - \frac{48 y^{3}}{3} + \frac{15 y^{2}}{2}}{2})}_{0}^{3} = [\frac{7 脳 3^{4}}{12} - \frac{42 脳 3^{3}}{9} + \frac{42 脳 3^{2}}{6} - \frac{35 脳 3}{3} + \frac{9 脳 3^{4}}{8} - \frac{48 脳 3^{3}}{6} + \frac{15 脳 3^{2}}{4}] - [0] = \frac{(567 脳 6) - (1134 脳 8) + (378 脳 12) - (105 脳 24) + (729 脳 9) - (1296 脳 12) + (135 脳 18)}{72} = \frac{3402 - 9072 + 4536 - 2520 + 6561 - 15552 + 2430}{72} = - \frac{10215}{72} = - \frac{1135}{8}

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Most popular questions from this chapter

Evaluate the sums in Exercises $23 鈥� 28$ .

{鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} (j - k)

Discuss the similarities and differences between the definition of the definite integral found in Chapter 4 and the definition of the double integral found in this section.

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

${鈭�}_{- 1}^{5} {鈭�}_{- 3}^{2} {鈭�}_{4}^{8} f (x, y, z) d z d x d y$

How many summands are in ${鈭�}_{i = i_{0}}^{l} {鈭�}_{j = j_{0}}^{m} {鈭�}_{k = k_{0}}^{n} k^{i} j^{k}$ ?

Earlier in this section, we showed that we could use Fubini鈥檚 theorem to evaluate the integral $鈭� {鈭�}_{R} x^{2} y d A$ and we showed that ${鈭�}_{2}^{5} {鈭�}_{1}^{3} x^{2} y d x d y = 91$ Now evaluate the double integral by evaluating the iterated integral ${鈭�}_{1}^{3} {鈭�}_{2}^{5} x^{2} y d y d x$ .

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