Use geometry or symmetry, or both, to evaluate the double integral. ∫∫DR2−x2−y2dA D is the disk with center the origin and radius R.

"Use geometry or symmetry, or both, to evaluate..." was answered by our community

he298c

2021-10-12

Use geometry or symmetry, or both, to evaluate the double integral.

\int \int_{D} \sqrt{R^{2} - x^{2} - y^{2}} d A

D is the disk with center the origin and radius R.

Pohanginah

Skilled2021-10-13Added 96 answers

\int \int_{D} \sqrt{R^{2} - x^{2} - y^{2}} d A

We can write

z = \sqrt{R^{2} - x^{2} - y^{2}}

which can become

z^{2} = R^{2} - x^{2} - y^{2}

x^{2} + y^{2} + z^{2} = R^{2}, z \geq 0

z must be

\geq 0

because the original function is the positive square root.

x^{2} + y^{2} + z^{2} = R^{2}

you will remember is the equation of a sphere, so with

z \geq 0

it is the upper half, a hemisphere.
We can find the volume using the sphere volume formula from geometry, just take half of it.

V = \frac{1}{2} \cdot \frac{4}{3} π R^{3} = \frac{2}{3} π R^{3}

Results:

\frac{2}{3} π R^{3}

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?