Respuesta :
The volume is obtained as 149.33 cu. units using double integral.
How to find the volume using double integral method?
A two-dimensional region can be integrated using double integrals. They enable us to, among other things, calculate the volume beneath a surface.
V= ∫Adx, or alternatively ∫Ady, where A denotes the usual disc's area. Hence, depending on the axis of revolution, r=f(x) or r=f(y). 2. The volume of the solid produced by a region under f(y) (to the left of f(y)) that is rotated about the y-axis and is bounded by the horizontal lines y=c and y=d.
Given equations are as follows.
z=x+y
[tex]x^2+y^2=16[/tex]
It can be modified as follows.
[tex]y^2=16-x^2[/tex]
So, value of x moves from 0 to 4 and y value moves from 0 to [tex]\sqrt{16-x^2}[/tex].
Integrate using Double integration using obtained limits to get volume.
[tex]V=\int_0^4 \int_0^{\sqrt{16-x^2}}(x+y) d y d x\\=\int_0^4\left[x y+\frac{y^2}{2}\right]_0^{\sqrt{16-x^2}} d x\\\begin{aligned}&=\int_0^4\left[x \sqrt{16-x^2}+\frac{16-x^2}{2}\right] d x \\&=\int_0^4x \sqrt{16-x^2} d x+\int_0^4 8 d x-\frac{1}{2} \int_0^4 x^2 d x\end{aligned}[/tex]
On integrating using proper substitution,
V=149.33 cu
So, the volume is obtained as 149.33 cu. units using double integral.
To know more about double integration:
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