Chapter 13: Q 17. (page 1014)

Use the results of Exercises 15 and 16 to find the area of the region $Ω$ shown in Exercise 13.

Short Answer

Expert verified

Area of region is $\frac{4}{3}$ units.

Step by step solution

Given Information

The region is bounded by curves $y = \sqrt{x} and y = \frac{1}{2} x$

Consideration

The region may be considered at type I or type II.

When it is expressed as type I region, it is bounded by $y = \frac{1}{2} x$ below and $y = \sqrt{x}$ above for all $x$ in interval $[0, 4]$

For type I region, $a = 0, b = 4$

Computation of area

Evaluation of Area,

The area is evaluated as $\iint_{Ω} d A = \int_{a}^{b g_{g_{1}} (x)} \int_{2}^{g_{2} (x)} d y d x$ ,

$\begin{array}{rcl} \int_{0}^{4} \int_{\frac{1}{2} x}^{\sqrt{x}} d y d x & = & \int_{0}^{4} [y]_{\frac{1}{2} x}^{\sqrt{x}} d x \\ = & {[\frac{2}{3} x^{3 / 2} - \frac{x^{2}}{4}]}_{0}^{4} \\ \int_{0}^{4} \int_{\frac{1}{2} x}^{\sqrt{x}} d y d x & = & \frac{2}{3} (8) - 4 \\ = & \frac{4}{3} \end{array}$

The required area $\frac{4}{3}$ .

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Use the results of Exercises 15 and 16 to find the area of the region $Ω$ shown in Exercise 13.

Short Answer

Step by step solution

Given Information

Consideration

Computation of area

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Use the results of Exercises 15 and 16 to find the area of the region Ωshown in Exercise 13.

Short Answer

Step by step solution

Given Information

Consideration

Computation of area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Geometry

Applied Mathematics

Mechanics Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Use the results of Exercises 15 and 16 to find the area of the region $Ω$ shown in Exercise 13.