Chapter 5: Q20P (page 241)

Above the rectangle with vertices (0,0), (0,1),(2,0),(2,1), and below the surface $z^{2} = 36 x^{2} (4 - x^{2})$

Short Answer

Expert verified

The required solution is 16.

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 rectangle with vertices at (0,0),(0,1),(2,0), and (2,1), and under the plane z = 8 - x + y .below the surface $z^{2} = 36 x^{2} (4 - x^{2}) .$

First, integrate over z:

$| = \int_{x = 0}^{2} \int_{y = 0}^{1} [\int_{z = 0}^{6 x \sqrt{4 - x^{2}}} d z] d x d y = \int_{0}^{1} d y \int_{0}^{2} d x 6 x \sqrt{4 - x^{2}}$

Further calculation of the volume of the bounded region

Now, integrate over x and y, such that,

$| = \int_{0}^{1} d y \int_{0}^{2} d x 6 x \sqrt{4 - x^{2}} = 6 \int_{0}^{2} \sqrt{4 - x^{2}} d (4 - x^{2}) (- \frac{1}{2}) = {- 3 \frac{2}{3} {(4 - x^{2})}^{3 / 2}}_{0}^{2} = - 2 (0 - 4^{3 / 2)} = 2 \times 2^{3} = 16$

Therefore, the value is 16

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Above the rectangle with vertices (0,0), (0,1),(2,0),(2,1), and below the surface $z^{2} = 36 x^{2} (4 - x^{2})$

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

One App. One Place for Learning.

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Above the rectangle with vertices (0,0), (0,1),(2,0),(2,1), and below the surface z2=36x2(4-x2)

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fundamentals of Physics

Medical Physics

Work Energy and Power

Electrostatics

Nuclear Physics

Particle Model of Matter

Study anywhere. Anytime. Across all devices.

Above the rectangle with vertices (0,0), (0,1),(2,0),(2,1), and below the surface $z^{2} = 36 x^{2} (4 - x^{2})$