∫ (from 0 to 4) of 4x-x^2 eching

$$\int _ { 0 } ^ { 4 } 4 x - x ^ { 2 }$$

Baholash

$$\frac{32}{3}\approx 10,666666667$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int 4x-x^{2}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 4x\mathrm{d}x+\int -x^{2}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$4\int x\mathrm{d}x-\int x^{2}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x\mathrm{d}x integralni \frac{x^{2}}{2} bilan almashtiring. 4 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$2x^{2}-\int x^{2}\mathrm{d}x$$

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. -1 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

$$2x^{2}-\frac{x^{3}}{3}$$

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

$$2\times 4^{2}-\frac{4^{3}}{3}-\left(2\times 0^{2}-\frac{0^{3}}{3}\right)$$

Qisqartirish.

$$\frac{32}{3}$$