March « 2008 « Solved Problems

Area, Volume and Perimeter

March 23, 2008

Consider the regions $\displaystyle \mathcal{R}_{1}\,\mathrm{and}\,\mathcal{R}_{2}$ enclosed by the curves $\displaystyle y=x^{3}+\frac{1}{2}$ and $\displaystyle x=-\left(y-\frac{1}{2}\right)^{2}$ . Set-up the integrals representing the following:

The perimeter of $\displaystyle \mathcal{R}_{2}$ .
The total area of the regions $\displaystyle \mathcal{R}_{1}$ and $\displaystyle \mathcal{R}_{2}$ .
The volume of the solid generated when $\displaystyle \mathcal{R}_{1}$ is rotated about the line $\displaystyle y=-\frac{1}{2}$ using the method of washers.
The volume of the solid generated when $\displaystyle \mathcal{R}_{2}$ is rotated about the line $x=-1$ using the method of cylindrical shells.

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5 Comments | Calculus | Tagged: arc length, area, cylindrical shells, volume, washers | Permalink
Posted by wMw

Largest Rectangle in Some Region in the First Quadrant

March 23, 2008

Find the area of the largest rectangle that can be inscribed in the region bounded by the curve $y=e^{-x}$ , the line $x=0$ and the positive $x$ -axis.

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Leave a Comment » | Calculus | Tagged: area, derivative tests, optimization, rectangle | Permalink
Posted by wMw

Normal Line Problem

March 23, 2008

Find the equation of the line normal to the curve $\displaystyle y=\tanh \left[\sin^{-1} \left( x-\sqrt{2}\right) \right]$ at the point where $\displaystyle x=\sqrt{2}$ .

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Leave a Comment » | Calculus | Tagged: chain rule, derivative, equation of line, normal line, slope, tangent line | Permalink
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Integrals Involving Transcendental Functions

March 23, 2008

Evaluate the following integrals.

$\displaystyle \int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\frac{\ln^{2}\left(\sin x\right)}{\tan x}\: dx$
$\displaystyle \int\frac{2^{x}\: dx}{\left(2^{x}+1\right)\sqrt{4^{x}+2^{x+1}-8}}$
$\displaystyle \int\frac{dx}{x+4\sqrt{x}+13}$

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4 Comments | Calculus | Tagged: definite integral, integration, inverse trigonometric functions, natural logarithmic function, substitution, transcendental functions | Permalink
Posted by wMw

Indeterminate Forms Involving Transcendental Functions

March 23, 2008

Find the limits.

$\displaystyle \lim_{x\rightarrow1^{+}}\left(\frac{1}{\ln x}-\frac{1}{\cosh^{-1}x}\right)$
$\displaystyle \lim_{x\rightarrow0^{+}}\left(e^{x}+\sinh x\right)^{{\displaystyle \cot x}}$

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Leave a Comment » | Calculus | Tagged: indeterminate forms, l'hospital's rule, limits, transcendental functions | Permalink
Posted by wMw

Logarithmic and Implicit Differentiation

March 23, 2008

Solve for $\displaystyle \frac{dy}{dx}$ .

$y=\displaystyle \sqrt[5]{\frac{4^{x}\coth\left(e^{-x}\right)}{\sin\left(x^{2}\ln x\right)}}$
$\displaystyle e^{y}=\cosh\left(e^{x}+\log_{3}y\right)-2^{{\displaystyle x^{2}}}$

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1 Comment | Calculus | Tagged: chain rule, derivatives, implicit differentiation, logarithmic differentiation, transcendental functions | Permalink
Posted by wMw

Continuity and L’Hospital’s Rule

March 22, 2008

Consider the function

$f(x) =\Biggl\{ \begin{array}{lcl} \cos x, & \mathrm{if} & x=0,\\ \displaystyle x \ln x (1-x) ^{-1}, & \mathrm{if} & 0<x<1, \\ -1, & \mathrm{if} & x=1. \end{array}$

Discuss the continuity of $f$ on $\left[0,1\right]$ .

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Leave a Comment » | Calculus | Tagged: continuity, indeterminate forms, l'hospital's rule | Permalink
Posted by wMw

Integrals of Some Transcendental Functions

March 21, 2008

Evaluate the following integrals:

1. $\displaystyle\int \frac{x^2}{1-\cosh^2 (x^3)} \ dx$
2. $\displaystyle\int 5^x \cot \big( 5^x \big) \ dx$
3. $\displaystyle\int \frac{dx}{x\sqrt{9x^4-16}}$
4. $\displaystyle\int\frac{x^2-4x+3}{x^2-4x+5} \ dx$