Chapter 5: Q23P (page 257)

In Problems 17 to 30, for the curve $y = \sqrt{x}$ , between role="math" localid="1658830478813" $x = 0$ and $x = 2$ ,
find:
The centroid of the surface area.

Short Answer

Expert verified

The centroid of the surface area $(x, y, z) = (\frac{149}{130}, 0, 0)$ .

Step by step solution

Definition of Centroid of a body

The point in a particular body at which the value of the volume is the same on all the sides is defined as the centroid of the body. If the mass is uniform in the body, then center of mass lies on the centeroid.

Representation of the area bounded by the curve

Draw the bounded region for the curve $y = \sqrt{x}$ , between $x = 0$ and $x = 2$ .

Calculation of the value of x

If the surface be rotated around the x - axis, then only $x$ remains, and $y = z = 0$ .

$x = \frac{1}{S} \int_{0}^{2} 2 πxf (x) dl = \frac{2 π}{S} \int_{0}^{2} xf (x) \sqrt{1 + {(f' (x))}^{2} dx} = \frac{2 π}{S} \int_{0}^{2} x \sqrt{x} \sqrt{1 + \frac{1}{4 x} dx} = \frac{2 π}{S} \int_{0}^{2} x \sqrt{x + \frac{1}{4} dx}$

Step 4`: Calculation of centroid of surface area

Evaluate the value, $S = \frac{13 π}{3}$ , so the integral is as follows,

$S = \frac{6}{13} [\int_{0}^{2} (x + \frac{1}{4}) \sqrt{1 + \frac{1}{4 x}} d (x + \frac{1}{4}) - \frac{1}{4} \int_{0}^{2} \sqrt{1 + \frac{1}{4 x}} d (x + \frac{1}{4})] = \frac{6}{13} {[\frac{2}{5} {(x + \frac{1}{4})}^{5 / 2} - \frac{1}{6} {(x + \frac{1}{4})}^{3 / 2}]}_{0}^{2} = \frac{149}{130}$

Thus, the centroid of the surface area $(x, y, z) = (\frac{149}{130}, 0, 0)$ .

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In Problems 17 to 30, for the curve $y = \sqrt{x}$ , between role="math" localid="1658830478813" $x = 0$ and $x = 2$ ,
find:
The centroid of the surface area.

Short Answer

Step by step solution

Definition of Centroid of a body

Representation of the area bounded by the curve

Calculation of the value of x

Step 4`: Calculation of centroid of surface area

One App. One Place for Learning.

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In Problems 17 to 30, for the curve y=x, between role="math" localid="1658830478813" x=0and x=2,find:The centroid of the surface area.

Short Answer

Step by step solution

Definition of Centroid of a body

Representation of the area bounded by the curve

Calculation of the value of x

Step 4`: Calculation of centroid of surface area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Fields in Physics

Turning Points in Physics

Atoms and Radioactivity

Nuclear Physics

Translational Dynamics

Waves Physics

Study anywhere. Anytime. Across all devices.

In Problems 17 to 30, for the curve $y = \sqrt{x}$ , between role="math" localid="1658830478813" $x = 0$ and $x = 2$ ,
find:
The centroid of the surface area.