Chapter 5: Q21P (page 247)

Above the triangle with vertices (0,0),(2,0), and (2,1), and below the paraboloid $z = 24 - x^{2} - y^{2}$ .

Short Answer

Expert verified

The required solution is $\frac{131}{6}$

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

Calculation of the volume under the curve

The volume above the triangle with vertices at (0,0),(2,0),(2,1) and below the paraboloid $z = 24 - x^{2} - y^{2}$ .

First, integrate over z:

$| = \int_{x = 0}^{2} \int_{y = 0}^{x / 2} [\int_{z = 0}^{24 - x^{2} - y^{2}} dz] dydx = \int_{0}^{2} dx \int_{0}^{x / 2} (24 - x^{2} - y^{2})$

Further calculation of the volume of the bounded region

Now, integrate over:

$| = \int_{0}^{2} dx {(24 y - x^{2} y - \frac{1}{3} y^{3})}_{0}^{x / 2} = \int_{0}^{2} (12 x - \frac{x^{3}}{2} - \frac{x^{3}}{24}) dx$

Further calculation of the volume of the bounded region

Now, integrate over x:

$| = {[6 x^{2} - \frac{x^{3}}{8} - \frac{x^{3}}{96}]}_{0}^{2} = \frac{131}{6}$

Therefore, the value is $\frac{131}{6}$

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Above the triangle with vertices (0,0),(2,0), and (2,1), and below the paraboloid $z = 24 - x^{2} - y^{2}$ .

Short Answer

Step by step solution

Definition of double integral

Calculation of the volume under the curve

Further calculation of the volume of the bounded region

Further calculation of the volume of the bounded region

One App. One Place for Learning.

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Above the triangle with vertices (0,0),(2,0), and (2,1), and below the paraboloid z=24-x2-y2.

Short Answer

Step by step solution

Definition of double integral

Calculation of the volume under the curve

Further calculation of the volume of the bounded region

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

Physics of Motion

Energy Physics

Torque and Rotational Motion

Medical Physics

Thermodynamics

Scientific Method Physics

Study anywhere. Anytime. Across all devices.

Above the triangle with vertices (0,0),(2,0), and (2,1), and below the paraboloid $z = 24 - x^{2} - y^{2}$ .