Question: "Find the volume of the solid in the..."

Jaya Legge

Answered question

2021-11-02

Find the volume of the solid in the first octant bounded by the coordinate planes, the plane x = 3, and the parabolic cylinder

z = 4 - y^{2}

Faiza Fuller

Skilled2021-11-03Added 108 answers

\int \int_{R} f (x, y) d A

To find the value of double integral, we must found the area of integration.
It is given:

f (x, y) = 4 - y^{2}

The domain is restricted by the area x=3.
Now, we can solve the integral:

\int_{0}^{2} \int_{0}^{3} (4 - y^{2}) d x d y = \int_{0}^{2} {[4 x - y^{2} x]}_{0}^{3} d y

= \int_{0}^{2} (12 - 3 y^{2}) d y

(Use

\int x^{a} d x = \frac{x^{a + 1}}{a + 1}

)

= {[12 y - y^{3}]}_{0}^{2}

=24-8
=16
Results: 16

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