Chapter 5: Q20P (page 257)

In Problems 17 to 30, for the curve $y = \sqrt{x}$ , between $x = 0$ and $x = 2$ ,
find:
The curved area of this solid.

Short Answer

Expert verified

The curved area of the solid is $\frac{13 π}{3}$

Step by step solution

Definition of double integral

Double integral of $f (x, y)$ over the area, A in the $(x, y)$ plane as the limit of this sum, and it is written as $\int \int_{A} f (x, y) d x d y$

Representation of the area bounded by the curve

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

Determination of the surface element of the solid generated when the area is revolved about the x-axis.

Determine the surface element from the figure.

$d S = 2 πf (x) dL = 2 πf (x) \sqrt{1 + {(f' (x))}^{2}} dx = 2 π \sqrt{x} \sqrt{1 + \frac{1}{4 x}} dx = 2 π \sqrt{x + \frac{1}{4}} dx$

Step 4: Determination of the total surface of the solid generated when the area is revolved about the axis.

Determine the total surface of the solid generated.

$S = 2 π \int_{0}^{2} \sqrt{x + \frac{1}{4}} d (x + \frac{1}{4}) {= \frac{4 π}{3} {(x + \frac{1}{4})}^{3 / 2}}_{0}^{2} = \frac{13 π}{3}$

Thus, the value of curved area of the solid is $\frac{13 π}{3}$ .

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In Problems 17 to 30, for the curve $y = \sqrt{x}$ , between $x = 0$ and $x = 2$ ,
find:
The curved area of this solid.

Short Answer

Step by step solution

Definition of double integral

Representation of the area bounded by the curve

Determination of the surface element of the solid generated when the area is revolved about the x-axis.

Step 4: Determination of the total surface of the solid generated when the area is revolved about the axis.

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In Problems 17 to 30, for the curve y=x, betweenx=0and x=2,find:The curved area of this solid.

Short Answer

Step by step solution

Definition of double integral

Representation of the area bounded by the curve

Determination of the surface element of the solid generated when the area is revolved about the x-axis.

Step 4: Determination of the total surface of the solid generated when the area is revolved about the axis.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Wave Optics

Fields in Physics

Atoms and Radioactivity

Scientific Method Physics

Work Energy and Power

Astrophysics

Study anywhere. Anytime. Across all devices.

In Problems 17 to 30, for the curve $y = \sqrt{x}$ , between $x = 0$ and $x = 2$ ,
find:
The curved area of this solid.