What is the integral from x^2 to 8-x^2 of y^2-x^2 ?

The integral from x^2 to 8-x^2 of y^2-x^2 is -8x^2+2x^4+(512-192x^2+24x^4-2x^6)/3

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integral from x^2 to 8-x^2 of y^2-x^2

\int_{x^{2}}^{8 - x^{2}} y^{2} - x^{2} d y

- 8 x^{2} + 2 x^{4} + \frac{512 - 192 x ^{2} + 24 x ^{4} - 2 x ^{6}}{3}

Solution steps

\int_{x^{2}}^{8 - x^{2}} y^{2} - x^{2} d y

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int_{x^{2}}^{8 - x^{2}} x^{2} d y + \int_{x^{2}}^{8 - x^{2}} y^{2} d y

\int_{x^{2}}^{8 - x^{2}} x^{2} d y = 8 x^{2} - 2 x^{4}

\int_{x^{2}}^{8 - x^{2}} y^{2} d y = \frac{512 - 192 x ^{2} + 24 x ^{4} - 2 x ^{6}}{3}

= - (8 x^{2} - 2 x^{4}) + \frac{512 - 192 x ^{2} + 24 x ^{4} - 2 x ^{6}}{3}

Simplify

= - 8 x^{2} + 2 x^{4} + \frac{512 - 192 x ^{2} + 24 x ^{4} - 2 x ^{6}}{3}

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What is the integral from x^2 to 8-x^2 of y^2-x^2 ?
The integral from x^2 to 8-x^2 of y^2-x^2 is -8x^2+2x^4+(512-192x^2+24x^4-2x^6)/3