Chapter 5: Q24P (page 247)

Under the surface z = 1 /(y+2) , and over the area bounded by and y=x $y^{2} + x = 2$ .

Short Answer

Expert verified

The required solution is $\frac{9}{2}$

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 $\int \int_{A} f (x, y) dxdy$

Drawing the area bounded by the curve inthe plane

The area is bounded by the graph y=x and $y^{2} + x = 2 in (x, y)$ the plane

Intersecting points identification

From the graph the intersections of the two curves are:

$y = - y^{2} y_{12} = \frac{- 1 \pm \sqrt{1 + 8}}{2} y_{12} = (- 2, 1)$

Calculation of the volume under the curve

First, integrate over z and x, such that,

$| = \int_{y = 2}^{1} d y \int_{x = y}^{2 - y^{2}} [\int_{z = 0}^{1 / (y + 2)} d z] d x = \int_{- 2}^{1} d y \int_{x}^{2 - y^{2}} \frac{d x}{y + 2} = \int_{- 2}^{1} {\frac{d x}{y + 2} x}_{y}^{2 - y^{2}} = \int_{- 2}^{1} \frac{d x}{y + 2} (2 - y^{2} - y)$

Further calculation of the volume of the bounded region

Now, integrate over y:

$| = \int_{- 2}^{1} \frac{d y}{y + 2} (y + 2) (y - 1) (- 1) = \int_{- 2}^{1} (y - 1) d y = {- [\frac{1}{2} y^{2} - y]}_{- 2}^{1} = \frac{9}{2}$

Therefore, the value is $\frac{9}{2}$

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Under the surface z = 1 /(y+2) , and over the area bounded by and y=x $y^{2} + x = 2$ .

Short Answer

Step by step solution

Definition of double integral

Drawing the area bounded by the curve inthe plane

Intersecting points identification

Calculation of the volume under the curve

Further calculation of the volume of the bounded region

One App. One Place for Learning.

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Under the surface z = 1 /(y+2) , and over the area bounded by and y=x y2+x=2.

Short Answer

Step by step solution

Definition of double integral

Drawing the area bounded by the curve inthe plane

Intersecting points identification

Calculation of the volume under the curve

Further calculation of the volume of the bounded region

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Work Energy and Power

Magnetism and Electromagnetic Induction

Scientific Method Physics

Famous Physicists

Kinematics Physics

Engineering Physics

Study anywhere. Anytime. Across all devices.

Under the surface z = 1 /(y+2) , and over the area bounded by and y=x $y^{2} + x = 2$ .