1. ## Volume of solid

Find the volume of a solid bounded by the paraboloid z = 4 – x2 – y2 and the xy plane.

2. Originally Posted by winsome
Find the volume of a solid bounded by the paraboloid z = 4 – x2 – y2 and the xy plane.
Have you drawn the region and determined your bounds for integration?

3. 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

4. Originally Posted by winsome
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:

$\displaystyle \displaystyle V = \int_{0}^{2 \pi} \int_0^2 (4 - r^2) r \, dr \, d\theta = ....$

5. Originally Posted by mr fantastic
$\displaystyle \displaystyle V = \int \int_{R_{xy}} 4 - x^2 - y^2 \, dy \, dx$ 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:

$\displaystyle \displaystyle V = \int_{0}^{2 \pi} \int_0^2 (4 - r^2) r \, dr \, d\theta = ....$

how to get limits 0 to 2pi and 0 to 2

6. Originally Posted by winsome
i m unable to show my next steps with integration sign, so please u show me its next steps after itegration between 0 to 2 pi and 0 to 2.
Integrate with respect to r. If you cannot do this then:

1. I don't know how you got to the stage of studying multivariable calculus.

2. I suggest you go back and review basic integration techniques.

Please show you work and say where you get stuck.

7. Originally Posted by mr fantastic
Integrate with respect to r. If you cannot do this then:

1. I don't know how you got to the stage of studying multivariable calculus.

2. I suggest you go back and review basic integration techniques.

Please show you work and say where you get stuck.
actually i want to know how to convert limits -2 to +2 and -sq root(4 - x^2) to +sq root(4 - x^2) in 0 to 2pi and 0 to 2 in our integration

8. Originally Posted by winsome
how to get limits 0 to 2pi and 0 to 2
Draw the circle! (Have you met polar coordinates before?)