1. By: Mohsin Tahir (GL) Waqas Akram Rao Arslan Ali Asghar Numan-ul-haq
VOLUME OF A SURFACE
By:
Mohsin Tahir (GL)
Waqas Akram
Rao Arslan
Ali Asghar
Numan-ul-haq

2. Surface The surface is the outside of anything.
The earth, a basketball, and even your body have a surface.
Surface

3. Volume
Volume is the measure of the amount of space inside of a solid figure, like a cube, ball, cylinder or pyramid.

4. The Volume Of A Cylinder.

5. The formula for the volume of a cylinder is:
V =  r 2 h
r = radius h = height.
Calculate the area of the circle:
A =  r 2
A = 3.14 x 2 x 2
A = cm2
Calculate the volume:
V =  r 2 x h
V = x 6
V = cm3

6. Sphere

7. Volume of a Cube

8. Volume of a cube = a × a × a = a³
where a is the length of each side of the cube.
Example:-
We want to find the volume of this cube in m3
According to formula:
v= 2m x 2m x 2m
v=8m
The volume of the cube is 8 m³ (8 cubic meters)

11.  Volume Under a Surface 
A double integral allows you to measure the volume under a surface as bounded by a rectangle.
Definite integrals provide a reliable way to measure the signed area between a function and the x-axis as bounded by any two values of x.
Similarly, a double integral allows you to measure the signed volume between a function z = f(x, y) and the xy-plane as bounded by any two values of x and any two values of y.

12. Double Integrals over Rectangles
double integrals by considering the simplest type of planar region, a rectangle. We consider a function ƒ(x, y) defined on a rectangular region R,
R : a ≤ x ≤ b, c ≤ y ≤ d
If the volume V of the solid that lies above the rectangle R and below the surface z = f(x, y) is:

13. Double Integrals as Volumes
dA= dy dx
dA= dx dy

14. Fubini’s Theorem for Calculating Double Integrals
Suppose that we wish to calculate the volume under the plane
Z = 4 - x - y
over the rectangular region R:
0 ≤ x ≤ 2 , 0 ≤ y ≤ 1
in the xy-plane. then the volume is:

15. 1
where A(x) is the cross-sectional area at x. For each value of x, we may calculate A(x) as the integral 2
which is the area under the curve Z = 4 - x - y in the plane of the cross-section at x. In calculating A(x), x is held fixed and the integration takes place with respect to y. Combining Equations (1) and (2), we see that the volume of the entire solid is:

16. If we just wanted to write a formula for the volume, without carrying out any of the integrations, we could write

17. Fubini’s Theorem
If ƒ(x, y) is continuous throughout the rectangular region then:
R : a ≤ x ≤ b, c ≤ y ≤ d

19. Examples to Finding the volume using Double integral

20. Q#1
Solution:

21. Q#2
Solution:

22. Q#3 Calculate the volume under the surface z=3 + X2 − 2y over the region D defined by 0 ≤ x ≤ 1 and −x ≤ y ≤ x.
Solution: The volume V is the double integral of z=3 + X2 − 2y over D.