Chapter 6: Q12P (page 314)

Evaluate each of the following integrals in the easiest way you can.
$\int (y^{2} - x^{2}) d x + (2 x y + 3) d y$ ,along the xaxis from (0,0) and localid="1659182150932" $(\sqrt 5, 0)$ then along
a circular arc from $(\sqrt 5, 0) to (1, 2) .$

Short Answer

Expert verified

The solution to this problem is $/ = \frac{29}{3}$

Step by step solution

Given Information.

The given information is $\int (y^{2} - x^{2}) d x + (2 x y + 3) d y$

Definition of Green’s Theorem.

The Green's theorem connects a line integral around a simple closed curve C to a double integral over the plane region D circumscribed by C in vector calculus. Stokes' theorem has a two-dimensional special case.

Find the solution.

Write the values of P and Q.

$P = y^{2} - x^{2} Q = 2 x y + 3$

Find the difference of their differentiation.

$\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 2 y - 2 y = 0$

The integral is in the form mentioned below.

$I = - \int_{1}^{0} (3 x^{2} + 8 x^{2} + 6) d x = \int_{1}^{0} (11 x^{2} + 6) d x = {[\frac{11}{3} x^{3} + 6 x]}_{0}^{1} = \frac{29}{3}$

Hence, the solution to this problem is $I = \frac{29}{3}$

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Evaluate each of the following integrals in the easiest way you can.
$\int (y^{2} - x^{2}) d x + (2 x y + 3) d y$ ,along the xaxis from (0,0) and localid="1659182150932" $(\sqrt 5, 0)$ then along
a circular arc from $(\sqrt 5, 0) to (1, 2) .$

Short Answer

Step by step solution

Given Information.

Definition of Green’s Theorem.

Find the solution.

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Evaluate each of the following integrals in the easiest way you can.∫(y2-x2)dx+(2xy+3)dy,along the xaxis from (0,0) and localid="1659182150932" (√5,0)then alonga circular arc from (√5,0)to(1,2).

Short Answer

Step by step solution

Given Information.

Definition of Green’s Theorem.

Find the solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Geometrical and Physical Optics

Oscillations

Fluids

Energy Physics

Atoms and Radioactivity

Study anywhere. Anytime. Across all devices.

Evaluate each of the following integrals in the easiest way you can.
$\int (y^{2} - x^{2}) d x + (2 x y + 3) d y$ ,along the xaxis from (0,0) and localid="1659182150932" $(\sqrt 5, 0)$ then along
a circular arc from $(\sqrt 5, 0) to (1, 2) .$