What is the integral from x^2+3y^2 to 8-x^2-y^2 of 1 ?

The integral from x^2+3y^2 to 8-x^2-y^2 of 1 is -2x^2-4y^2+8

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integral from x^2+3y^2 to 8-x^2-y^2 of 1

\int_{x^{2} + 3 y^{2}}^{8 - x^{2} - y^{2}} 1 d z

- 2 x^{2} - 4 y^{2} + 8

Solution steps

\int_{x^{2} + 3 y^{2}}^{8 - x^{2} - y^{2}} 1 d z

Integral of a constant:

\int a d x = a x

= [1 \cdot z]_{x^{2} + 3 y^{2}}^{8 - x^{2} - y^{2}}

Simplify

= [z]_{x^{2} + 3 y^{2}}^{8 - x^{2} - y^{2}}

Compute the boundaries:

- 2 x^{2} - 4 y^{2} + 8

= - 2 x^{2} - 4 y^{2} + 8

\int_{1}^{4} (x - 2)^{4} d x

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\int_{- 2}^{3} x + 2 d x

\int_{1}^{4} (\frac{3}{x ^{2}} - 1) d x

\int_{\frac{3}{2}}^{2} \frac{3}{8} x^{2} d x

What is the integral from x^2+3y^2 to 8-x^2-y^2 of 1 ?
The integral from x^2+3y^2 to 8-x^2-y^2 of 1 is -2x^2-4y^2+8