∫ _{R_{1}}^R_{2}(x^2+y^2) eching

$$\int _ { R _ { 1 } } ^ { R _ { 2 } } ( x ^ { 2 } + y ^ { 2 } )$$

Baholash

$$\frac{\left(R_{2}-R_{1}\right)\left(3y^{2}+R_{1}R_{2}+R_{1}^{2}+R_{2}^{2}\right)}{3}$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int x^{2}+y^{2}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int x^{2}\mathrm{d}x+\int y^{2}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{2}\mathrm{d}x integralni \frac{x^{3}}{3} bilan almashtiring.

$$\frac{x^{3}}{3}+\int y^{2}\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, y^{2} integralini toping.

$$\frac{x^{3}}{3}+y^{2}x$$

Xos integral bu integral hisoblashning yuqori chegarasida hisoblangan ifodaning boshlangʻich holatidan chiqarib tashlagan holda integral hisoblashning quyi chegarasida hisoblangan ifodaning boshlangʻich holatidir.

$$\frac{R_{2}^{3}}{3}+y^{2}R_{2}-\left(\frac{R_{1}^{3}}{3}+y^{2}R_{1}\right)$$

Qisqartirish.

$$\frac{\left(-R_{1}+R_{2}\right)\left(3y^{2}+R_{1}^{2}+R_{1}R_{2}+R_{2}^{2}\right)}{3}$$

y ga nisbatan hosilani topish

$$2y\left(R_{2}-R_{1}\right)$$