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:

  1. The perimeter of \displaystyle \mathcal{R}_{2}.
  2. The total area of the regions \displaystyle \mathcal{R}_{1} and \displaystyle \mathcal{R}_{2}.
  3. 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.
  4. The volume of the solid generated when \displaystyle \mathcal{R}_{2} is rotated about the line x=-1 using the method of cylindrical shells.

Read the rest of this entry »

5 Comments | Calculus | Tagged: , , , , | Permalink
Posted by wMw

Related Rates

March 23, 2008

A particle moves along the curve y=\ln x so that its abscissa is increasing at a rate of 2 units per second. At what rate is the particle moving away from the origin as it passes through the point (e,1)?

Read the rest of this entry »

Leave a Comment » | Calculus | Tagged: , , | 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.

Read the rest of this entry »

Leave a Comment » | Calculus | Tagged: , , , | 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}.

Read the rest of this entry »

Leave a Comment » | Calculus | Tagged: , , , , , | Permalink
Posted by wMw

Integrals Involving Transcendental Functions

March 23, 2008

Evaluate the following integrals.

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

Read the rest of this entry »

4 Comments | Calculus | Tagged: , , , , , | Permalink
Posted by wMw

Indeterminate Forms Involving Transcendental Functions

March 23, 2008

Find the limits.

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

Read the rest of this entry »

Leave a Comment » | Calculus | Tagged: , , , | Permalink
Posted by wMw

Logarithmic and Implicit Differentiation

March 23, 2008

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

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

Read the rest of this entry »

1 Comment | Calculus | Tagged: , , , , | Permalink
Posted by wMw