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Chapter 6: Problem 30

x=1-t^{2}, y=2 t, 0 \leq t \leq 1

Short Answer

Expert verified
The Cartesian equation is \(x = 1 - \frac{y^2}{4}\) for \(0 \leq y \leq 2\).

Step by step solution

01

Parametric Equations

The given problem involves parametric equations where the variables \(x\) and \(y\) are expressed as functions of a parameter \(t\). Given equations are \(x = 1 - t^2\) and \(y = 2t\).
02

Determine Range of \(t\)

The parameter \(t\) has a defined range of \(0 \leq t \leq 1\). This range will help in determining how \(x\) and \(y\) vary as \(t\) changes.
03

Eliminate Parameter \(t\) to Find Cartesian Equation

To eliminate the parameter, solve one of the parametric equations for \(t\). From \(y = 2t\), we have \(t = \frac{y}{2}\). Substitute \(t = \frac{y}{2}\) into \(x = 1 - t^2\), giving the equation \(x = 1 - \left(\frac{y}{2}\right)^2\).
04

Simplify the Cartesian Equation

Simplify the equation \(x = 1 - \frac{y^2}{4}\) to make it a neat Cartesian form. This equation describes the curve without the parameter \(t\).
05

Determine the Values for \(y\) and \(x\) by Range of \(t\)

From given \(0 \leq t \leq 1\): when \(t=0\), \(y=0\) and \(x=1\), and when \(t=1\), \(y=2\) and \(x=0\). So as \(t\) varies from 0 to 1, \(y\) varies from 0 to 2, and \(x\) decreases from 1 to 0.

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Key Concepts

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

Cartesian Equation
Parametric equations can be transformed into a single equation in terms of the variables involved, known as a Cartesian equation. In our given exercise, we start with the parametric equations: \(x = 1 - t^2\) and \(y = 2t\). The goal is to eliminate the parameter \(t\) to combine these into a Cartesian equation solely in terms of \(x\) and \(y\). This results in a more familiar form that can easily be graphed, understood, and analyzed without involving the parameter.
Parameter Elimination
Parameter elimination is the process of removing the parameter to express the relationship between \(x\) and \(y\) directly. In our example, we have \(y = 2t\) which can be rearranged to express \(t\) as \(t = \frac{y}{2}\).
Substituting \(t = \frac{y}{2}\) in the equation \(x = 1 - t^2\), we get \(x = 1 - \left(\frac{y}{2}\right)^2\).
This simplifies to the Cartesian equation \(x = 1 - \frac{y^2}{4}\). This equation now links \(x\) and \(y\) directly, without the parameter \(t\), giving a single equation to describe the curve.
Range of Parameter
The range of the parameter is crucial in determining the values that the variables \(x\) and \(y\) can take. For our exercise, \(t\) is confined to the range \(0 \leq t \leq 1\). This range affects how \(x\) and \(y\) change:
  • When \(t=0\), substituting in, \(y=0\) and \(x=1\).
  • When \(t=1\), substituting in, \(y=2\) and \(x=0\).
This range helps define the segment of the curve that the parametric equations trace out, thus revealing how the curve progresses as \(t\) varies.
Graphing Curves
Graphing the curves derived from parametric equations involves plotting the values of \(x\) and \(y\) as \(t\) changes within its specified range. Once we have the Cartesian equation \(x = 1 - \frac{y^2}{4}\), this curve can be plotted without needing to vary \(t\) explicitly.
The given range for \(t\) from 0 to 1 shows that \(y\) takes values from 0 to 2, and consequently, \(x\) decreases from 1 to 0. This is a typical methodogram to localize the curve segments of the equation within a cartesian plane.
With this method, any point on the curve corresponds to a specific \(t\) value, which provides a complete picture of the curve's path as represented by the equations.

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Most popular questions from this chapter

Sketch the graph of the four-cusped hypocycloid \(x=a \sin ^{3} t, y=a \cos ^{3} t, 0 \leq t \leq 2 \pi\), and find its length. Hint: By symmetry, you can quadruple the length of the first quadrant portion.

In Problems 1-12, find the volume of the solid generated when the region \(R\) bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps. (a) Sketch the region \(R\). (b) Show a typical rectangular slice properly labeled. (c) Write a formula for the approximate volume of the shell generated by this slice. (d) Set up the corresponding integral. (e) Evaluate this integral. \(x=y^{2}, y=1, x=0 ;\) about the \(x\)-axis

In Problems 1-12, find the volume of the solid generated when the region \(R\) bounded by the given curves is revolved about the indicated axis. Do this by performing the following steps. (a) Sketch the region \(R\). (b) Show a typical rectangular slice properly labeled. (c) Write a formula for the approximate volume of the shell generated by this slice. (d) Set up the corresponding integral. (e) Evaluate this integral. \(x=\sqrt{2 y}+1, y=2, x=0, y=0 ;\) about the line \(y=3\)

In Problems 11-30, sketch the region bounded by the graphs of the given equations, show a typical slice, approximate its area, set up an integral, and calculate the area of the region. Make an estimate of the area to confirm your answer: \(y=\sqrt[3]{x}, y=0\), between \(x=-2\) and \(x=2\)

Find the area of the region trapped between \(y=x e^{-x^{2}}\) and \(y=x / 4\). Hint: There are two separate regions.
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