/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 11 In Exercises 1 through 20 , eval... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 20: Problem 11

In Exercises 1 through 20 , evaluate the line integral over the given curve. \(\int_{C}\left(x y+x^{2}\right) d x+x^{2} d y ; C:\) the parabola \(y=2 x^{2}\) from the origin to the point \((1,2)\).

Short Answer

Expert verified
\(\frac{11}{6}\)

Step by step solution

01

- Parametrize the curve

The curve given is a parabola described by the equation \(y=2x^{2}\). Parametrize the curve by setting \(x=t\) for \(t \in [0,1]\). Thus, \(x=t\) and \(y=2t^2\).
02

- Compute \(dx\) and \(dy\)

Differentiate the parametrization with respect to \(t\). Find \(dx/dt = 1\) so \(dx = dt\) and \(dy/dt = 4t\) so \(dy = 4t \, dt\).
03

- Substitute into the integral

Substitute \(x\), \(y\), \(dx\), and \(dy\) into the integral: \[\int_{0}^{1} \left( t \(2t^2\) + t^{2} \right) dt + t^{2} (4t) dt = \int_{0}^{1} \left( 2t^3 + t^2 + 4t^3 \right) dt \]
04

- Simplify the integrand

Simplify the integrand to get: \[\int_{0}^{1} 6t^3 + t^2 \, dt\right) \]
05

- Integrate

Integrate each term separately: \[\int_{0}^{1} 6t^3 + t^2 \, dt = \left[ 6 \frac{t^4}{4} + \frac{t^3}{3} \right]_{0}^{1} = \left[ \frac{3}{2}t^4 + \frac{1}{3}t^3 \right]_{0}^{1} = \left( \frac{3}{2} + \frac{1}{3} \right) - (0) = \frac{11}{6} \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parametrize the Curve
To evaluate a line integral, the first step is to parametrize the curve. This means describing the curve using a parameter, often denoted as t. For the given problem, the curve is a parabola, which can be expressed as the equation \(y=2x^2\). To parametrize, we set \(x=t\) where \(t\) ranges from 0 to 1. This helps us because now we can write \(x\) and \(y\) in terms of \(t\), so we have \(x=t\) and \(y=2t^2\). This transformation simplifies the process of evaluating line integrals.
Differentiation
Once the curve is parametrized, the next step is to find the differentials \(dx\) and \(dy\). Since we have \(x=t\), differentiating \(x\) with respect to \(t\) gives us \frac{dx}{dt}=1\(, so \)dx = dt\(. For \)y=2t^2\(, differentiating with respect to \)t\( gives us \frac{dy}{dt}=4t\), so \(dy=4t \, dt\). Differentiation is crucial because it allows us to substitute \(dx\) and \(dy\) in the integral, making it easier to work with the parameters rather than the original variables, especially in more complex integrals.
Integration
After substituting \(x\), \(y\), \(dx\), and \(dy\) into the integral, we proceed to the integration step. Here, we combine and simplify the terms under the integral to make the integral easier to solve. For the problem at hand, simplifying gives us \(\frac{11}{6}\) as the answer. Integrating involves finding the antiderivatives of the function with respect to the parameter \(t\) over the specified interval, followed by evaluating these antiderivatives at the boundaries of the interval. This is a fundamental step in solving line integrals as it allows us to find the total accumulated value along the curve.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

An equation of the surface of a mountain is \(z=1200-3 x^{2}-2 y^{2}\), where distance is measured in feet, the \(x\) axis points to the east, and the \(y\) axis points to the north. A mountain climber is at the point corresponding to \((-10,5,850)\). (a) What is the direction of steepest ascent? (b) If the climber moves in the east direction, is he ascending or descending, and what is his rate? (c) If the climber moves in the southwest direction, is he ascending or descending, and what is his rate? (d) In what direction is he traveling a level path?

In Exercises 21 through 34 , find the total work done in moving an object along the given arc \(C\) if the motion is caused by the given force field. Assume the arc is measured in inches and the force is measured in pounds. \(\mathbf{F}(\boldsymbol{x}, y, z)=x \mathbf{i}+y \mathfrak{j}+(y z-x) \mathbf{k} ; C: \mathbf{R}(t)=2 t \mathbf{i}+t^{2} \mathbf{j}+4 t^{3} \mathbf{k}, 0 \leq t \leq 1\)

A manufacturing plant has two classifications for its workers, A and B. Class A workers earn \(\$ 14\) per run, and class B workers earn \(\$ 13\) per run. For a certain production run, it is determined that in addition to the salaries of the workers, if \(x\) class A workers and \(y\) class B workers are used, the number of dollars in the cost of the run is \(y^{3}+x^{2}-8 x y+600\). How many workers of each class should be used so that the cost of the run is a minimum if at least three workers of each class are required for a run?

\(\int_{C}(\sin y \sinh x+\cos y \cosh x) d x+(\cos y \cosh x-\sin y \sinh x) d y ; A\) is \((1,0)\) and \(B\) is \((2, \pi) ;\) Exercise 4

The demand equations for two commodities that are produced by a monopolist are $$ x=6-2 p+q \quad \text { and } \quad y=7+p-q $$ where \(100 x\) is the quantity of the first commodity demanded if the price is \(p\) dollars per unit and \(100 y\) is the quantity of the second commodity demanded if the price is \(q\) dollars per unit. Show that the two commodities are substitutes. If it costs \(\$ 2\) to produce each unit of the first commodity and \(\$ 3\) to produce each unit of the second commodity, find the quantities demanded and the prices of the two commodities in order to have the greatest profit. Take \(p\) and \(q\) as the independent variables.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.