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Chapter 13: Q 70. (page 1068)

Let $b$ be a constant. Prove that the equation of the plane $y = b$ is $r = b c s c θ$ in cylindrical coordinates.

Short Answer

Expert verified

Conversion of equation is done using $x = r \cos θ, y = r \sin θ, z = z$ .

Step by step solution

Given Information

Equation of plane in rectangular coordinates is given by $y = b$ ( $b$ is constant)

Simplification

We know that

$x = r \cos θ, y = r \sin θ, z = z$

For the transformation, use $y = r \sin θ$ in $y = b$

$r \sin θ = b$

$r = \frac{b}{\sin θ}$

$\Rightarrow r = b \csc θ$

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Most popular questions from this chapter

Evaluate each of the double integrals in Exercises 37–54 as iterated integrals.

$\int \int_{R} x^{2} \cos (x y) d A, w h e r e R = {(x, y) | 0 \leq x \leq π a n d 0 \leq y \leq 1}$

Describe the three-dimensional region expressed in each iterated integral in Exercises 35–44.

$\int_{- 3}^{3} \int_{- \sqrt{9 - x^{2}}}^{\sqrt{9 - x^{2}}} \int_{- 3}^{3} f (x, y, z) d z d y d x$

Evaluate the iterated integral :

$\int_{0}^{π / 2} \int_{0}^{\cos y} \int_{0}^{\sin y} (2 x + y) d z d x d y$

Evaluate Each of the integrals in exercises 33-36 as an iterated integral and then compare your answer with thoise you found in exercise 29-32

$\int \int_{R} x y d A W h e r e R = {(x, y) | 0 \leq x \leq 2, 1 \leq y \leq 4}$

Identify the quantities determined by the integral expressions in Exercises 19–24. If x, y, and z are all measured in centimeters and ρ(x, y,z) is a density function in grams per cubic centimeter on the three-dimensional region , give the units of the expression.

$∭_{Ω} x ρ (x, y, z) d V$

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Let bbe a constant. Prove that the equation of the plane y=bis r=bcscθin cylindrical coordinates.

Short Answer

Step by step solution

Given Information

Simplification

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Most popular questions from this chapter

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Let $b$ be a constant. Prove that the equation of the plane $y = b$ is $r = b c s c θ$ in cylindrical coordinates.