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Chapter 5: Q33P (page 257)

Verify that (3.10) gives the same result as (3.8).

Short Answer

Expert verified

The given equation is verified.

Step by step solution

01

Representation of the equations

Write equation 3.10.

$V = \int_{x = 0}^{1} \int_{_{(} z = - x^{2}}^{x^{2}} \int_{y - \sqrt{(x^{4} - z^{2})}}^{\sqrt{(x^{4} - z^{2})}} = d y d z d x$

Write equation 3.8.

$V = \int_{0}^{1} π y^{2} d x = \int_{0}^{1} π x^{4} d x = \frac{π}{5}$

02

Verification of the given equation

Simplify equation 3.10.

$V = \int_{0}^{1} d x \int_{- x^{2}}^{x^{2}} d z \int_{- \sqrt{x^{4} - z^{2}}}^{\sqrt{x^{4} - z^{2}}} d y = \int_{0}^{1} d x \int_{- x^{2}}^{x^{2}} 2 \sqrt{x^{4} - z^{2}} = 2 \int_{0}^{1} d x \int_{- x^{2}}^{x^{2}} \sqrt{x^{4} - z^{2}} d z$

It is known that $z = x^{2} s i n u, a n d d z = x^{2} c o s u d u$ Substitute the values in the above expression.

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Most popular questions from this chapter

Under the surface z = y(x+2) , and over the area bounded by $x + y = 0, y = 1, y = \sqrt{x}$ .

In Problems 17 to 30, for the curve $y = \sqrt{x}$ , between $x = 0$ and $x = 2$ ,

find:

The area under the curve.

(a) Write a triple integral in cylindrical coordinates for the volume of the part of a ball between two parallel planes which intersect the ball.

(b) Evaluate the integral in (a). Warning hint: Do the r and $θ$ integrals first.

(c) Find the centroid of this volume.

$\int_{x = 0}^{4} \int_{y = 0}^{\sqrt{x}} \sqrt[y]{x} dydx$

Find the Jacobians $\partial (x, y) / \partial (u, y)$ of the given transformations from the variables x,y to variables u,v :

$x = \frac{1}{2} (u^{2} - v^{2}) y = u v$

( u and v are called parabolic cylinder coordinates)

See all solutions

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