Chapter 5: Q33P (page 248)

A lamina covering the quarter disk $x^{2} + y^{2} \leq 4, x > 0, y > 0$ has (area) density . Find the mass of the lamina.

Short Answer

Expert verified

The mass of the lamina is, $\frac{16}{3} \cdot$

Step by step solution

Definition of double integral and mass formula

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

Using the double integral the mass of the lamina: $M = \int \int_{A} ρdA .$

Calculation of the mass of the lamina

To get the mass of the lamina, the area of integration is the quarter-circle with a radius $R = 2$ .

The integrals:

$M = \int \int_{A} ρdA = \int \int_{A} (x + y) dxdy$

Further calculation of the mass of the lamina

Change into the polar coordinate system,

$M = \int_{0}^{2} \int_{0}^{π / 2} r (\cos θ + \sin θ) rdrdθ = \int_{0}^{2} r^{2} dr \int_{0}^{π / 2} (\cos θ + \sin θ) dθ = {\frac{8}{3} (sinθ - cosθ)}_{0}^{π / 2} = \frac{8}{3} (1 - 0 - 0 + 1) = \frac{16}{3}$

Therefore, the mass of the lamina is, $\frac{16}{3}$ .

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A lamina covering the quarter disk $x^{2} + y^{2} \leq 4, x > 0, y > 0$ has (area) density . Find the mass of the lamina.

Short Answer

Step by step solution

Definition of double integral and mass formula

Calculation of the mass of the lamina

Further calculation of the mass of the lamina

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A lamina covering the quarter disk x2+y2≤4,x>0,y>0 has (area) density . Find the mass of the lamina.

Short Answer

Step by step solution

Definition of double integral and mass formula

Calculation of the mass of the lamina

Further calculation of the mass of the lamina

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Conservation of Energy and Momentum

Magnetism and Electromagnetic Induction

Nuclear Physics

Further Mechanics and Thermal Physics

Measurements

Study anywhere. Anytime. Across all devices.

A lamina covering the quarter disk $x^{2} + y^{2} \leq 4, x > 0, y > 0$ has (area) density . Find the mass of the lamina.