engineering math3

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Applied Engineering Mathematics

IntegrationIntegration

There is no simplegeometric formula forcalculating the areas ofshapes having curvedboundaries l ike theregion R.








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Integration tipsIntegration tips




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Simplifying the integrand

We consider changes of variable, x, later under various substitution methods, buthere we will have a first look at what we might be able to do with the integrand insome simple cases. In general f (x) may be a polynomial or rational function, trig orhyperbolic, logor exponential, or any combination of these. We would first look at f(x) to see if it can be simplified or rearranged to our advantage. we concentrate onthe simplest kinds of rearrangements. For example:





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Linear substitution in integration







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So, greater care is needed with other types of substitutions.

These questions all involve linear substitutions. In no case is it necessary to doanything to the function before substitution.









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The du= f (x)/ dx substitution yada

Integration by substitutionIntegration by substitution






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2. zm





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Integration by partsIntegration by parts




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Applied Engineering Mathematics Applied Engineering Mathematics

Multiple IntegralsMultiple Integrals

The process of integration for one variable can be extended to the functions of morethan one variable. The generalization of definite integrals is known as multiple integral.

Consider the region R in the x, y plane weassume that R is a closed, bounded regionin t he x , y plane, by the curve y = f1(x), y =f2(x) and the lines x = a, x = b. Let us laydown a rectangulargrid on R consisting of afinite number of lines parallel to thecoordinate axes. The N rectangles lyingentirely within R (the shaded ones in thefigure). Let (xr, yr) be an arbitrarily selected

point in the rth partition rectangle for each r= 1, 2, ..., N. Then denoting the area xr yr = Sr

Thus, the total sum of areas


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Let the maximum linear dimensions of each portion of areas approach zero, and nincreases indefinitely then the sum SNwill approach a limit, namely the double integral

and the value of this limit is given by



Double integration for polar curves or Integrals in Polar Coordinates





Double Integrals as Volumes

When (x, y) is a positive function over a rectangular region R in the xy-plane, we mayinterpret the double integral of over R as the volume of the 3-dimensional solid regionover thexy-plane bounded below by R and above by the surface z=f(x,y).


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