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Sagot :
The volume of the given solid, with the boundaries, is given as follows:
661.5 cubic units.
How to obtain the volume of the solid?
The volume of the solid is obtained using a double integral.
The equation for the solid is given as follows:
7x + 9y + z = 63.
z = 63 - 7x - 9y.
The coordinate planes bound the plane, hence:
- y varies between 0 and 7 - 7x/9.
- x varies between 0 and 9.
Hence the double integral that is used to obtain the volume of the solid is given as follows:
[tex]V = \int_{0}^{9}\int_{0}^{7 - \frac{7x}{9}} 63 - 7x - 9y dydx[/tex]
The inner integral is given as follows:
[tex]I = \int_{0}^{7 - \frac{7x}{9}} 63 - 7x - 9y dy[/tex]
Which, applying the Fundamental Theorem of Calculus, has the result given as follows:
[tex]I = -98x + \frac{49x^2}{9} - \frac{(7x + 63)^2}{18} + 441[/tex]
Then the volume of the solid is given by the outer integral as follows:
[tex]V = \int_{0}^{9} \left(-98x + \frac{49x^2}{9} - \frac{(7x + 63)^2}{18} + 441\right) dx[/tex]
Which has a numeric value of:
661.5 cubic units.
More can be learned about integrals and volumes at https://brainly.com/question/25870210
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