Find the volume of a solid bounded by the paraboloid z = 4 – x2 – y2 and the xy plane.
Yes i have solved it as below but not sure that i m correct and now stuck to get it complete solution. Please help me to get its complete solution
from to given equation 4 – x2 – y2 = 0
Solve for ‘y’ i.e.
y2 = 4 – x2
y =
Also for y = 0,
4 – x2 = 0
x = ±2
Hence, Volume is given by
V =
Let substitute,
x2 + y2 = r2 _______(2)
x = r sin qÞ dx = r cos q dq
y = r cos qÞ dy = –r sin q dq
$\displaystyle \displaystyle V = \int \int_{R_{xy}} 4 - x^2 - y^2 \, dy \, dx$ where $\displaystyle R_{xy}$ is the region in the xy-plane defined by the interior of the circle $\displaystyle x^2 + y^2 = 4$.
Switch to polar coordinates:
$\displaystyle \displaystyle V = \int_{0}^{2 \pi} \int_0^2 (4 - r^2) r \, dr \, d\theta = ....$