Chapter 6: Q11P (page 314)

Evaluate each of the following integrals in the easiest way you can.
$\int_{C} (x s i n x - y) d x + (x - y^{2}) d y$ ,where Cis the triangle in the (x, y) plane with
vertices (0,0), (1,1), and (2,0).

Short Answer

Expert verified

The solution to this problem is $I = 2$

Step by step solution

Given Information

The given information is $\int_{C} (x \sin x - y) d x + (x - y^{2}) d y .$

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.

$P = x \sin x - y Q = x - y^{2}$

Find the difference of their differentiation

$\begin{array}{rcl} \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} & = & 1 + 1 \\ = & 2 \end{array}$

The integral is in the form mentioned below.

$\begin{array}{rcl} I & = & \iint d y d x \\ = & 2 \times a r e a o f t h e t r i a n g l e \\ = & 2 \end{array}$

Hence, the solution to this problem is . $I = 2$

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Evaluate each of the following integrals in the easiest way you can.
$\int_{C} (x s i n x - y) d x + (x - y^{2}) d y$ ,where Cis the triangle in the (x, y) plane with
vertices (0,0), (1,1), and (2,0).

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.∫C(xsinx-y)dx+(x-y2)dy,where Cis the triangle in the (x, y) plane withvertices (0,0), (1,1), and (2,0).

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

Translational Dynamics

Fluids

Rotational Dynamics

Electromagnetism

Physics of Motion

Circular Motion and Gravitation

Study anywhere. Anytime. Across all devices.

Evaluate each of the following integrals in the easiest way you can.
$\int_{C} (x s i n x - y) d x + (x - y^{2}) d y$ ,where Cis the triangle in the (x, y) plane with
vertices (0,0), (1,1), and (2,0).