# Multiple Integral

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multiple integrals

. chapter we extend the idea of a definite integral to double and triple integrals of functions of two or three variables. These. able to consider in Chapters 6 andWe also use double integrals to calculate probabilities when two random variables are involved. We will see that polar coordinates are useful in computing double integrals over some types of regions. In a similar way, we. and spherical coordinates-that greatly simplify the computation of triple integrals over certain commonly occurring solid regions.multiple integrals

There is no general rule for predicting whose order of integration will be better than the other order like Example 9.’s Theorem 9.1.3 like the above example, the integration boundary should be properly set up: Example 9.1.5. x = −y + 1 to 1 − y 2 . Thus √ the iterated integral can be written as 0 −y+1 f (x, y. from y = −x + 1 to 1 − x2 . Thus the √ iterated integral can be written asmultiple integration

. in two dimensions, leading to what is called an “iterated integral”. In section 17.3 we shall give a more formal deﬁnition of the double integral, and then see that its computation uses the technique of. under the graph z f x y is the double integral of f over R, denoted (17.1)multiple integrals

. we are willing to give. But the properties of the integral need and deserve attention:Linearity: jj(f + g)dA = jj.multiple integrals

. a parallel between the de…nite integral and double integrals. In the case of the de…nite integral, the width of each subinterval. their sides are approachingRemark 314 The de…nition of the integral does not require f (x; y) to be positive onWhen f (x; y) 0 on R, the integral is exactly the volume of the solidRemark 315 We can approximate double integrals by using double Riemann sums. In the next section, we.
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