∫ (from 0 to 2 cos θ) of r ^ 2dr eching

$$\int _ { 0 } ^ { 2 \cos \theta } r ^ { 2 } d r$$

Baholash

$$\frac{8\left(\cos(\theta )\right)^{3}}{3}$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int r^{2}\mathrm{d}r$$

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

$$\frac{r^{3}}{3}$$

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

$$\frac{1}{3}\times \left(2\cos(\theta )\right)^{3}-\frac{0^{3}}{3}$$

Qisqartirish.

$$\frac{8\left(\cos(\theta )\right)^{3}}{3}$$

θ ga nisbatan hosilani topish

$$-8\sin(\theta )\left(\cos(\theta )\right)^{2}$$