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Calculus

I'm having trouble with the following problem:

Find the volume of the solid generated by revolving the region about the given line. The region is in the first quadrant bounded above by the line y= sqrt 2, below by the curve y=secxtanx, and on the left by the y-axis. Rotate the region about the line y=sqrt 2.

The problem is supposed to be done using the disk method.
So far I know that I need to find the point of intersection betwen y= sqrt 2 and y=secxtanx
secxtanx= sqrt 2
I'm not sure how to do the math to find that, but when I used my calculator to approximate the intersection I got .78539816. I know that this will be my upper bound, and the lower bound is zero. I also know the general formula for volume is
v= pi (integral from a to b) [f(x)^2]dx
I also know that the answer is supposed to be pi(pi/2 + 2sqrt2 + 11/3). However, I don't know how to actually solve the problem.

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Jenna

  1. Yes, you have done your homework, and you are not far from finishing it.

    First, the disk method is well described in the following article. In the later part of the solution, if you have problems, you can refer to it.
    http://www.vias.org/calculus/06_applications_of_the_integral_02_02.html

    A sketch of the volume to be found can be found at the following link:
    http://img85.imageshack.us/img85/1486/jenna.png

    Your expression for finding the volume by the disk method is correct. All that is missing is the function for the "radius" of the elemental disks, and the limits a and b.

    We know from the question description that the lower limit is 0. The upper limit is as you have found it, π/4.
    It can be found as follows:
    tan(x)/cos(x) = sqrt(2)
    By expanding tan(x) into sin/cos, and applying the identity sin²+cos&sup2 = 1 you will end up with a quadratic equation in sin(x),
    sqrt(2)sin²(x)+sin(x)-sqrt(2)=0
    of which the positive solution is x=π/4.

    The function f(x) representing the radius of the solid of rotation expressed as a function of x is evident from the sketch:
    f(x) = sqrt(2)-tan(x)/cos(x)
    So proceed with the definite integral and you will succeed in finding the correct answer.

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    MathMate

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