Chapter 5: Q6P (page 247)

$\int_{y - 1}^{2} \int_{x - \sqrt{y}}^{y^{2}} x d x d y$

Short Answer

Expert verified

The required solution is $\int_{y - 1}^{2} \int_{x - \sqrt{y}}^{y^{2}} xdxdy \frac{47}{20}$ .

Step by step solution

Definition of Definite Integral

The antiderivative and indefinite integrals of a function are closely connected to the definite integral. The indefinite integral, if it exists, has a real numerical value, whereas the latter two represent an unlimited number of functions that differ only by a constant.

Finding the Integral

Consider the given integral,

$\int_{y - 1}^{2} \int_{x - \sqrt{y}}^{y^{2}} xdxdy = \int_{y - 1}^{2} [\frac{1}{2} x^{2}_{\sqrt{y}}^{y^{2}}] d y = \frac{1}{2} \int_{1}^{2} (y^{4} - y) d y = \frac{1}{10} y^{5}_{1}^{2} - \frac{1}{4} y^{2}_{1}^{2} = \frac{1}{10} (32 - 1) - \frac{1}{4} (4 - 1) = \frac{62}{10} - \frac{15}{20} = \frac{47}{20}$

Therefore, the solution is $\int_{y - 1}^{2} \int_{x - \sqrt{y}}^{y^{2}} xdxdy = \frac{47}{20}$ .

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$\int_{y - 1}^{2} \int_{x - \sqrt{y}}^{y^{2}} x d x d y$

Short Answer

Step by step solution

Definition of Definite Integral

Finding the Integral

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∫y-12∫x-yy2xdxdy

Short Answer

Step by step solution

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Finding the Integral

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Most popular questions from this chapter

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$\int_{y - 1}^{2} \int_{x - \sqrt{y}}^{y^{2}} x d x d y$