Chapter 6: Q1P (page 334)

Do case (b) of Example $1$ above.

Short Answer

Expert verified

The value of the integral is $I = - 3 {πa}^{2}$ .

Step by step solution

Given Information

The movement in the counter clockwise direction on the $XY$ plane is $dr = - dx \hat{i} + dy \hat{j}$ .

Definition of Integration

In mathematics, integration is a process of finding a function $g (X)$ whose derivative, $Dg (X)$ , is equal to a given function $f (X)$ .

Use the curve equation and integrate

The movement in the counter clockwise direction on the XY plane is $dr = - dx \hat{i} + dy \hat{j}$ .

So, $Vdr = - 4 ydx + xdy$ .

The equation of the curve is $x^{2} + y^{2} = a^{2}$

This implies:

$y = \sqrt{a^{2} - x^{2}}$

$dy = \frac{- x}{\sqrt{a^{2} - x^{2}}}$

The value of the integral is:

$I = 4 \int_{0}^{a} [- a \sqrt{a^{2} - x^{2}} + \frac{x^{2}}{\sqrt{a^{2} - x^{2}}}] dx$

$= 4 \int_{0}^{a} \frac{5 x^{2} - 4 a^{2}}{\sqrt{a^{2} - x^{2}}} dx$

Change the variables.

$x = asin θ$

$dx = acos θdθ$

Change the limit.

$0 \to 0$ and $a \to \frac{π}{2}$

Hence, the value of the integral is $I = - 3 {πa}^{2}$

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Do case (b) of Example $1$ above.

Short Answer

Step by step solution

Given Information

Definition of Integration

Use the curve equation and integrate

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Do case (b) of Example 1above.

Short Answer

Step by step solution

Given Information

Definition of Integration

Use the curve equation and integrate

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Famous Physicists

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Study anywhere. Anytime. Across all devices.

Do case (b) of Example $1$ above.