Whakaoti i te ∫ (from 0 to 3) of (9-x^2)^2

$$\int _ { 0 } ^ { 3 } ( 9 - x ^ { 2 } ) ^ { 2 }$$

Aromātai

$$\frac{648}{5}=129.6$$

Ngā Upane Otinga

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(9-x^{2}\right)^{2}.

$$\int _{0}^{3}81-18x^{2}+\left(x^{2}\right)^{2}\mathrm{d}x$$

Hei hiki pū ki tētahi pū anō, me whakarea ngā taupū. Me whakarea te 2 me te 2 kia riro ai te 4.

$$\int _{0}^{3}81-18x^{2}+x^{4}\mathrm{d}x$$

Aromātaitia te tau tōpū tautuhi-kore i te tuatahi.

$$\int 81-18x^{2}+x^{4}\mathrm{d}x$$

Kōmitimititia te kīanga tapeke mā te kīanga.

$$\int 81\mathrm{d}x+\int -18x^{2}\mathrm{d}x+\int x^{4}\mathrm{d}x$$

Whakatauwehea te pūmau i ēnei kīanga katoa.

$$\int 81\mathrm{d}x-18\int x^{2}\mathrm{d}x+\int x^{4}\mathrm{d}x$$

Kimihia te tau tōpū o 81 mā te whakamahi i te ture mō te ripanga o ngā tau tōpū pātahi \int a\mathrm{d}x=ax.

$$81x-18\int x^{2}\mathrm{d}x+\int x^{4}\mathrm{d}x$$

Nā te mea \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} mō te k\neq -1, me whakakapi \int x^{2}\mathrm{d}x ki te \frac{x^{3}}{3}. Whakareatia -18 ki te \frac{x^{3}}{3}.

$$81x-6x^{3}+\int x^{4}\mathrm{d}x$$

Nā te mea \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} mō te k\neq -1, me whakakapi \int x^{4}\mathrm{d}x ki te \frac{x^{5}}{5}.

$$81x-6x^{3}+\frac{x^{5}}{5}$$

Ko te tau tōpū tautuhi ko te pārōnaki kōaro o te kīanga i aromātaitia i te tepe tōrunga o te pāwhaitua, tangohia te pārōnaki kōaro i aromātaitia i te tepe tōraro o te pāwhaitua.

$$\frac{3^{5}}{5}-6\times 3^{3}+81\times 3-\left(\frac{0^{5}}{5}-6\times 0^{3}+81\times 0\right)$$

Whakarūnātia.

$$\frac{648}{5}$$