Chapter 5: Q18P (page 247)

$\iint y^{- 1 / 2} d x d y$ over the area bounded by $y = x^{2}, x + y = 2$ , and theaxis.

Short Answer

Expert verified

The required solution is 1.438.

Step by step solution

Definition of double integral

The double integral of $f (x, y)$ over the areaA in the $(x, y)$ plane as the limit of this sum, and we write it as $\int \int_{A} f (x, y) d x d y$ .

Drawing the area bounded by the curve

The area is bounded by the curve $y = x^{2}, x + y = 2$ and y- axis.

Intersecting points identification

From the figure the curves the intersection points:

$x^{2} = 2 - x x^{2} + x - 2 = 0 x_{1, 2} = \frac{- 1 \pm \sqrt{1 + 8}}{2} x_{1, 2 = (1, - 2)}$

The integration limit will be x=0 from to x=1.

Calculation of the area under the curve

First, integrate over y:

$\int \int_{A} \frac{d x d y}{\sqrt{y}} = \int_{x = 0}^{1} [\int_{y = x^{2}}^{2 - x} \frac{d y}{\sqrt{y}}] d x = \int_{0}^{1} (2 {\sqrt{y}}_{x^{2}}^{2 - x}) d x = 2 \int_{0}^{1} (\sqrt{2 - x} - x) d x$

Further calculation of the area of the bounded region

Now, integrate over x :

$2 \int_{0}^{1} (\sqrt{2 - x} - x) d x = 2 - [\int_{0}^{1} \sqrt{2 -} x d (2 - x) - {\frac{1}{2} x^{2}}_{0}^{1}] = - \frac{4}{3} (2 - x) {^{3 / 2}}_{0}^{1} - 1 = - \frac{7}{3} + \frac{\sqrt[8]{2}}{3} \approx 1.438$

Therefore, the value is 1.438.

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$\iint y^{- 1 / 2} d x d y$ over the area bounded by $y = x^{2}, x + y = 2$ , and theaxis.

Short Answer

Step by step solution

Definition of double integral

Drawing the area bounded by the curve

Intersecting points identification

Calculation of the area under the curve

Further calculation of the area of the bounded region

One App. One Place for Learning.

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∬y-1/2dxdy over the area bounded by y=x2,x+y=2, and theaxis.

Short Answer

Step by step solution

Definition of double integral

Drawing the area bounded by the curve

Intersecting points identification

Calculation of the area under the curve

Further calculation of the area of the bounded region

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Classical Mechanics

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Study anywhere. Anytime. Across all devices.

$\iint y^{- 1 / 2} d x d y$ over the area bounded by $y = x^{2}, x + y = 2$ , and theaxis.