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Chapter 6: Problem 6

Let \(f(x)=1-x^{2}\) and \(g(x)=a x^{2}-a,\) where \(a\) is an unknown positive real number. For what value(s) of \(a\) is the area between the curves \(f\) and \(g\) equal to \(2 ?\)

Short Answer

Expert verified
The value of \(a\) is 1.

Step by step solution

01

Find the intersection points of the curves

To find the intersection points of the curves, set the equations equal to each other: \(1 - x^2 = a x^2 - a\). Rearrange the equation to: \(1 - x^2 - a x^2 + a = 0\) Combine like terms: \(a - (1 + a)x^2 = 0\). Factor out \(x^2\): \(a = (1 + a)x^2\). Thus, the solutions for \(x\) are \(x = \pm\sqrt{\frac{a}{1 + a}}\).
02

Set up the integral for the area between the curves

The area between the curves from \(x = -\sqrt{\frac{a}{1 + a}}\) to \(x = \sqrt{\frac{a}{1 + a}}\) is given by the integral: \[\int_{-\sqrt{\frac{a}{1 + a}}}^{\sqrt{\frac{a}{1 + a}}} [f(x) - g(x)]\,dx\]. Substitute the given functions: \[\int_{-\sqrt{\frac{a}{1 + a}}}^{\sqrt{\frac{a}{1 + a}}} [(1 - x^2) - (a x^2 - a)]\,dx\]. Simplify the integrand: \[\int_{-\sqrt{\frac{a}{1 + a}}}^{\sqrt{\frac{a}{1 + a}}} [1 - x^2 - a x^2 + a]\,dx\] = \[\int_{-\sqrt{\frac{a}{1 + a}}}^{\sqrt{\frac{a}{1 + a}}} [1 + a - (1 + a)x^2]\,dx\].
03

Evaluate the integral

Evaluate the integral: \[\int_{-\sqrt{\frac{a}{1 + a}}}^{\sqrt{\frac{a}{1 + a}}} [1 + a - (1 + a)x^2]\,dx\]. This separates into two integrals: \[(1 + a)\int_{-\sqrt{\frac{a}{1 + a}}}^{\sqrt{\frac{a}{1 + a}}} 1 \,dx - (1 + a)\int_{-\sqrt{\frac{a}{1 + a}}}^{\sqrt{\frac{a}{1 + a}}} x^2 \,dx\]. Evaluate the first integral: \[(1 + a)\left[2\sqrt{\frac{a}{1 + a}}\right]\]. Evaluate the second integral using the fact that it is an even function over symmetric limits, which results in zero: \[(1 + a)\int_{-\sqrt{\frac{a}{1 + a}}}^{\sqrt{\frac{a}{1 + a}}} x^2 \,dx = (1 + a) \frac{2}{3} \left(\sqrt{\frac{a}{1 + a}}\right)^3\].
04

Solve for the value of \(a\)

We equate the evaluated integral to 2: \[2(1 + a)\sqrt{\frac{a}{1 + a}} - \frac{2}{3}(1 + a)\left(\sqrt{\frac{a}{1 + a}}\right)^3 = 2\]. First term is reduced to: \[2\sqrt{a(1 + a)}\]. To solve: \[2\sqrt{a(1 + a)} - \frac{2}{3} a\sqrt{\frac{a}{1 + a}} = 2\]. By simplifying, it leads to a quadratic equation solving for \(a\): \[a = 1\].

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Key Concepts

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

definite integrals
Definite integrals play a crucial role in calculating the area between curves in calculus. The definite integral of a function over an interval essentially sums the tiny slices of area under the curve within that interval.

To find the area between two curves, say \(f(x)\) and \(g(x)\), you integrate the difference \([f(x) - g(x)]\) over the interval where they intersect.

An essential part of this process is setting up the integral with the correct limits of integration, which are usually the points where \(f(x)\) and \(g(x)\) intersect. This method helps in determining the exact area enclosed between the curves.
intersection points
Finding intersection points is essential for calculating the area between curves. These points determine the limits of integration. You find the intersection points by setting the functions equal to each other and solving for \(x\).

For instance, given \(f(x) = 1 - x^2\) and \(g(x) = ax^2 - a\), you find the intersection points by solving:

\[1 - x^2 = ax^2 - a\]

Rearranging the above equation, you get:

\[1 - ax^2 + a - 1 = (1 + a)x^2\]

From here, solving for \(x\) results in \(x = \pm\sqrt{\frac{a}{1+a}}\). These solutions give the points where the two curves intersect and thus set the limits for the definite integral.
quadratic equations
Quadratic equations often appear in problems involving curves, especially parabolas. They are written in the form \(ax^2 + bx + c = 0\). The solutions to quadratic equations, known as roots, can be found using various methods such as factoring, completing the square, or the quadratic formula.

For our purposes, when solving \(1 - x^2 = ax^2 - a\), we essentially rearrange it into the standard quadratic format to find the intersection points. By solving:

\[ax^2 - (1+a)x^2 + 1 - a = 0\]

we derived that \(x = \pm\sqrt{\frac{a}{1+a}}\).

Understanding quadratic equations is vital as the parabolic shapes of \(f(x)\) and \(g(x)\) make calculating the intersection points and areas straightforward when using algebraic methods.

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Most popular questions from this chapter

Consider the curve given by \(y=f(x)=2 x e^{-1.25 x}+(30-x) e^{-0.25(30-x)}\). a. Plot this curve in the window \(x=0 \ldots 30, y=0 \ldots 3\) (with constrained scaling so the units on the \(x\) and \(y\) axis are equal), and use it to generate a solid of revolution about the \(x\) -axis. Explain why this curve could generate a reasonable model of a baseball bat. b. Let \(x\) and \(y\) be measured in inches. Find the total volume of the baseball bat generated by revolving the given curve about the \(x\) -axis. Include units on your answer c. Suppose that the baseball bat has constant weight density, and that the weight density is 0.6 ounces per cubic inch. Find the total weight of the bat whose volume you found in (b). d. Because the baseball bat does not have constant cross-sectional area, we see that the amount of weight concentrated at a location \(x\) along the bat is determined by the volume of a slice at location \(x\). Explain why we can think about the function \(\rho(x)=\) \(0.6 \pi f(x)^{2}\) (where \(f\) is the function given at the start of the problem) as being the weight density function for how the weight of the baseball bat is distributed from \(x=0\) to \(x=30\) e. Compute the center of mass of the baseball bat.

A fuel oil tank is an upright cylinder, buried so that its circular top is 10 feet beneath ground level. The tank has a radius of 7 feet and is 21 feet high, although the current oil level is only 17 feet deep. Calculate the work required to pump all of the oil to the surface. Oil weighs \(50 \mathrm{lb} / \mathrm{ft}^{3}\). Work = ___________.

A rod has length 3 meters. At a distance \(x\) meters from its left end, the density of the rod is given by \(\delta(x)=4+5 x \mathrm{~g} / \mathrm{m}\). (a) Complete the Riemann sum for the total mass of the rod (use \(D x\) in place of \(\Delta x\) ): mass \(=\Sigma\) ___________. (b) Convert the Riemann sum to an integral and find the exact mass. \(\operatorname{mass}=\) ___________.

Calculate the integral, if it converges. If it diverges, enter diverges for your answer. \(\int_{-\infty}^{1} \frac{e^{5 x}}{1+e^{5 x}} d x=\) ___________.

Sometimes we may encounter an improper integral for which we cannot easily evaluate the limit of the corresponding proper integrals. For instance, consider \(\int_{1}^{\infty} \frac{1}{1+x^{3}} d x .\) While it is hard (or perhaps impossible) to find an antiderivative for \(\frac{1}{1+x^{3}},\) we can still determine whether or not the improper integral converges or diverges by comparison to a simpler one. Observe that for all \(x>0,1+x^{3}>x^{3},\) and therefore $$ \frac{1}{1+x^{3}}<\frac{1}{x^{3}} $$ It therefore follows that $$ \int_{1}^{b} \frac{1}{1+x^{3}} d x<\int_{1}^{b} \frac{1}{x^{3}} d x $$ for every \(b>1\). If we let \(b \rightarrow \infty\) so as to consider the two improper integrals \(\int_{1}^{\infty} \frac{1}{1+x^{3}} d x\) and \(\int_{1}^{\infty} \frac{1}{x^{3}} d x,\) we know that the larger of the two improper integrals converges. And thus, since the smaller one lies below a convergent integral, it follows that the smaller one must converge, too. In particular, \(\int_{1}^{\infty} \frac{1}{1+x^{3}} d x\) must converge, even though we never explicitly evaluated the corresponding limit of proper integrals. We use this idea and similar ones in the exercises that follow. a. Explain why \(x^{2}+x+1>x^{2}\) for all \(x \geq 1,\) and hence show that \(\int_{1}^{\infty} \frac{1}{x^{2}+x+1} d x\) converges by comparison to \(\int_{1}^{\infty} \frac{1}{x^{2}} d x\). b. Observe that for each \(x>1, \ln (x)2\). Why must it be true that \(\int_{2}^{b} \frac{1}{\ln (x)} d x\) diverges? c. Explain why \(\sqrt{\frac{x^{4}+1}{x^{4}}}>1\) for all \(x>1\). Then, determine whether or not the improper integral $$ \int_{1}^{\infty} \frac{1}{x} \cdot \sqrt{\frac{x^{4}+1}{x^{4}}} d x $$ converges or diverges.
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