x = tan t and then you're basically doing
The curve has equation
a] use the substitution to evaluate the arc length of correct to 2 decimal places.
b] The arc of joining the points and is rotated through four right angles about the x-axis. Find the area of the curved surface generated correct to 2 decimal places.
I've done part a] - got 2.96.
Part b] is giving me significant trouble though. I write the integral as
and from here, I've tried a variety of techniques - I made the term into and used the rule but the answer was wrong, and I tried the same hyperbolic substitution, which also gave me a wrong answer. I know this because the answer is written down on a sheet with no working to show it was obtained. Please could you suggest a suitable way to approach the question? Any help is greatly appreciated
Trigonometric substitution - Wikipedia, the free encyclopedia
gives the strategies. Pic on its way.
Just in case a picture helps...
... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).
The general drift is...
And the rest...
... is the drift for the parts process, except we've zoomed in on the embedded chain rule. For see http://en.wikipedia.org/wiki/Integral_of_secant_cubed
Plug in the limits and use a triangle diagram or http://staff.jccc.net/swilson/trig/compositions.htm to simplify the trig.
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!