Answer to Calculus 1 example. Here is a photo

Tpsaini13

Answered question

2022-04-01

alenahelenash

Expert2023-04-26Added 556 answers

To set up the double integral for the volume of the solid bounded by the paraboloid

z = 100 - 4 x^{2} - 25 y^{2}

and the coordinate planes in the first octant, we first need to determine the limits of integration for

x

y

, and

z

.
Since the paraboloid is symmetric about the

z

-axis, we can consider only the first octant, where

x \geq 0

and

y \geq 0

.
The paraboloid intersects the

x y

-plane when

z = 0

, so we have:

100 - 4 x^{2} - 25 y^{2} = 0

Solving for

y

, we get:

y = \frac{2}{5} \sqrt{25 x^{2} - 100}

Thus, for a given value of

x

y

ranges from

0

\frac{2}{5} \sqrt{25 x^{2} - 100}

.
Similarly, for a given

x

and

y

z

ranges from

0

100 - 4 x^{2} - 25 y^{2}

.
Therefore, the double integral that represents the volume of the solid is:

\int_{V} 1 d V = \int_{0}^{\sqrt{10}} \int_{0}^{\frac{2}{5} \sqrt{25 x^{2} - 100}} \int_{0}^{100 - 4 x^{2} - 25 y^{2}} d z d y d x

Here, the limits of integration for

x

are from

0

\sqrt{10}

because the paraboloid intersects the

x z

-plane when

y = 0

x = \sqrt{10} / 2

.
Note that we do not evaluate this integral. This is just the setup for finding the volume of the solid.

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