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Revision:Further Integration
TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > Further Integration
Contents |
Hyperbolic Standard Integrals
Inverse Trig and Hyperbolic Integrals
Reduction Formula
The Reduction Formula is used when integrals can be reduced by generalising the integral for a function to the powers of n. From this you can substitute a value for n and evaluate a definite or indefinite integral relatively easy. They are especially useful when integrating large powers of trigonometric functions.
For example :
Find 
take 
The 
refers to the integral where there is a power of n involved.
Evaluating this by integration by parts:
So
Notice that 
becomes 
From this, we can find
by saying "let n = 3".
Therefore,
Therefore,
Therefore,
This can be used for 
and other trigonometric functions.
Length of a Curve
Arc Length AB = ![\displaystyle \int_{x_a}^{x_b} \left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{1}{2}} \hspace5 dx](http://www.thestudentroom.co.uk/latexrender/pictures/33/333062ae4c82ace6b1118243dcebb1de.png "\displaystyle \int_{x_a}^{x_b} \left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{1}{2}} \hspace5 dx")
Arc Length AB = ![\displaystyle \int_{x_a}^{x_b} \left[1+\left(\frac{dx}{dy}\right)^2\right]^{\frac{1}{2}} \hspace5 dx](http://www.thestudentroom.co.uk/latexrender/pictures/1d/1ddfaa5ac38e49f7ba28d3d28e5a6490.png "\displaystyle \int_{x_a}^{x_b} \left[1+\left(\frac{dx}{dy}\right)^2\right]^{\frac{1}{2}} \hspace5 dx")
Arc Length AB = ![\displaystyle \int_{x_a}^{x_b} \left[\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2\right]^{\frac{1}{2}} \hspace5 dt](http://www.thestudentroom.co.uk/latexrender/pictures/12/12c37291b77d880bcd20623b7be0581e.png "\displaystyle \int_{x_a}^{x_b} \left[\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2\right]^{\frac{1}{2}} \hspace5 dt")
Area of Surface of Revolution
When rotated around the x axis:
Surface Area = ![\displaystyle \int_{x_a}^{x_b} \displaystyle 2\pi y \left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{1}{2}} \hspace5 dx](http://www.thestudentroom.co.uk/latexrender/pictures/65/65b0c147cd23641fb5bbed86b9f8a3b8.png "\displaystyle \int_{x_a}^{x_b} \displaystyle 2\pi y \left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{1}{2}} \hspace5 dx")
When using parametric coordinates (for rotation around the x axis):
Surface Area = ![\displaystyle \int_{x_a}^{x_b} 2\pi y \left[\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2\right]^{\frac{1}{2}} \hspace5 dt](http://www.thestudentroom.co.uk/latexrender/pictures/09/09541a9cbbd05d49a490515d761f1840.png "\displaystyle \int_{x_a}^{x_b} 2\pi y \left[\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2\right]^{\frac{1}{2}} \hspace5 dt")
When rotated around the y axis:
Surface Area = ![\displaystyle \int_{x_a}^{x_b} \displaystyle 2\pi x \left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{1}{2}} \hspace5 dx](http://www.thestudentroom.co.uk/latexrender/pictures/2f/2f65f7f09dc96c792cc6f101246b3f7d.png "\displaystyle \int_{x_a}^{x_b} \displaystyle 2\pi x \left[1+\left(\frac{dy}{dx}\right)^2\right]^{\frac{1}{2}} \hspace5 dx")
