# 微积分作业代写calclulus代考| Integration over complex domains

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## 微积分作业代写calclulus代考|Integration over complex domains

and do not depend on $y$. Only in the case of rectangular domains will the limits of both the inner and outer integrals be constants!

For an $x$-simple domain, with $D=\left{(x, y): c \leq y \leq d, h_{1}(y) \leq x \leq h_{2}(y)\right}$, we get the analogous result
$$\iint_{D} f(x, y) \mathrm{d} A=\int_{C}^{d} \mathrm{~d} y \int_{h_{1}(y)}^{h_{2}(y)} f(x, y) \mathrm{d} x .$$
Once again, the limits of the inner integral depend on the outer integral variable; the limits of the inner integral here depend on $y$ and do not depend on $x$.

Through a very simple development we have arrived at very natural generalizations of iterated integrals over rectangles. Moreover, in the process we have done away with the extensions we used in this development.

The reader should now bear two things in mind. First, the order in which the iterated integrals are to be performed must be strictly adhered to. Second, interchanging the order will always involve a change in the limits. This is illustrated in Example $4.2$ wherein a double integral is evaluated in two ways. The reader should note the limits on the two inner integrals and how they come about (see the vertical and horizontal bars in Figure 4.14).

## 微积分作业代写calclulus代考|A y-simple domain in a rectangle R

Third, we bring these ideas together to arrive at a strategy for evaluating integrals over non-rectangular domains. We demonstrate this with a $y$-simple domain (Fig. 4.13).

By construction we have the iterated integral of $\hat{f}$ over $R=[a, b] \times[c, d] \supset D$,
$$\iint_{D} f(x, y) \mathrm{d} A=\iint_{R} \hat{f}(x, y) \mathrm{d} A=\int_{a}^{b} \mathrm{~d} x \int_{c}^{d} \hat{f}(x, y) \mathrm{d} y$$
However, for every value of the outer integral variable $x, \hat{f}=0$ outside the interval $g_{1}(x) \leq y \leq g_{2}(x)$, and $\hat{f}=f$ in the interior of that interval. Hence,
$$\iint_{D} f(x, y) \mathrm{d} A=\underbrace{\int_{a}^{b} \mathrm{~d} x \int_{g_{1}(x)}^{g_{2}(x)} f(x, y) \mathrm{d} y}_{\text {iterated integral of } f \text { over } D}$$
We can now invoke the contrast alluded to earlier regarding the variable dependence of the limits of the inner integral. For all cases of non-rectangular domains, the limits of the inner integral will be functions of the outer integral variable. In the above example, the limits on the inner integral depend on $x$

## 微积分作业代写calclulus代考|Integration over complex domains

∬DF(X,和)d一种=∫Cd d和∫H1(和)H2(和)F(X,和)dX.

## 微积分作业代写calclulus代考|A y-simple domain in a rectangle R

∬DF(X,和)d一种=∬RF^(X,和)d一种=∫一种b dX∫CdF^(X,和)d和

∬DF(X,和)d一种=∫一种b dX∫G1(X)G2(X)F(X,和)d和⏟的迭代积分 F 超过 D