Har bir mintaqa bir xil egri chiziq bilan chegaralangan bo'lishiga qaramay, biz \(D\) ni to'liq tavsiflaydigan yagona chegara to'plamini ololmaymiz. Yuqori mintaqada \(x = {y^3}\) o'ng chegara va pastki mintaqada \(x = {y^3}\) chap chegara hisoblanadi.

Shuning uchun, har bir mintaqa uchun alohida chegaralar to'plami kerak bo'ladi va shuning uchun biz integralni quyidagicha ajratishimiz kerak.

\[\iint\limits_{D}{{{{\bf{e}}^{\,{y^{\,4}}}}\,dA}} = \iint\limits_{{{D_{\,1}}}}{{{{\bf{e}}^{\,{y^{\,4}}}}\,dA}} + \iint\limits_{{{D_{\,2}}}}{{{{\bf{e}}^{\,{y^{\,4}}}}\,dA}}\]

Umid qilamanki, biz avval \(x\) ni ikkala integral bilan birlashtirishimiz kerakligi aniq. Shunday qilib, bu erda har bir integral uchun chegaralar mavjud.

\[\begin{array}{*{20}{c}}\begin{array}{c}{D_{\,1}}\\ \,0 \le y \le 1\\ 0 \le x \le {y^3}\end{array}&{\hspace{0.5in}}&\begin{array}{c}{D_{\,2}}\\ - 1 \le y \le 0\\ {y^3} \le x \le 0\end{array}\end{array}\]

U holda integrallar,

\[\iint\limits_{D}{{{{\bf{e}}^{\,{y^{\,4}}}}\,dA}} = \int_{0}^{1}{{\int_{0}^{{{y^3}}}{{{{\bf{e}}^{\,{y^{\,4}}}}\,dx}}\,dy}} + \int_{{ - 1}}^{0}{{\int_{{{y^3}}}^{0}{{{{\bf{e}}^{\,{y^{\,4}}}}\,dx}}\,dy}}\]

Endi integrallarni bajarishdan boshqa ko'p narsa qilish kerak emas. Mana ikkalasi uchun \(x\) integratsiyasi.

\[\begin{align*}\iint\limits_{D}{{{{\bf{e}}^{\,{y^{\,4}}}}\,dA}} & = \int_{0}^{1}{{\left. {\left( {x{{\bf{e}}^{\,{y^{\,4}}}}} \right)} \right|_0^{{y^3}}\,dy}} + \int_{{ - 1}}^{0}{{\left. {\left( {x{{\bf{e}}^{\,{y^{\,4}}}}} \right)} \right|_{{y^3}}^0\,dy}}\\ & = \int_{0}^{1}{{{y^3}{{\bf{e}}^{\,{y^{\,4}}}}\,dy}} + \int_{{ - 1}}^{0}{{ - {y^3}{{\bf{e}}^{\,{y^{\,4}}}}\,dy}}\end{align*}\]

Va nihoyat, ikkala integral uchun \(y\) integratsiyasi.

\[\begin{align*}\iint\limits_{D}{{{{\bf{e}}^{\,{y^{\,4}}}}\,dA}} & = \int_{0}^{1}{{{y^3}{{\bf{e}}^{\,{y^{\,4}}}}\,dy}} + \int_{{ - 1}}^{0}{{ - {y^3}{{\bf{e}}^{\,{y^{\,4}}}}\,dy}}\\ & = \left. {\left( {\frac{1}{4}{{\bf{e}}^{\,{y^{\,4}}}}} \right)} \right|_0^1 + \left. {\left( { - \frac{1}{4}{{\bf{e}}^{\,{y^{\,4}}}}} \right)} \right|_{ - 1}^0\\ & = \,\frac{1}{4}\left( {{\bf{e}} - 1} \right) + \frac{1}{4}\left( { - 1 + {\bf{e}}} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{1}{2}\left( {{\bf{e}} - 1} \right) = 0.8591}}\end{align*}\]

Har doim mintaqadagi har bir integral bitta integral bilan bajarilishini kutmang. Vaqti-vaqti bilan siz integralni bo'lishingiz va alohida kichik mintaqalar bo'yicha haqiqiy integratsiyani amalga oshirishingiz kerak bo'ladi. Bu holda, bu juda aniq edi, lekin ba'zida muammoga duch kelmaguningizcha va uni kichik hududlarda qilish osonroq bo'lishini tushunmaguningizcha, bu unchalik aniq bo'lmasligi mumkin.