volume of a surface by: mohsin tahir (gl) waqas akram rao arslan ali asghar numan-ul-haq
TRANSCRIPT
VOLUME OF A SURFACE
By:
Mohsin Tahir (GL)
Waqas Akram
Rao Arslan
Ali Asghar
Numan-ul-haq
Surface
The surface is the outside of anything.
The earth, a basketball, and even your body have a surface.
Surface
Volume
Volume is the measure of the amount of space inside of a solid figure, like a cube, ball, cylinder or pyramid.
The Volume Of A Cylinder.
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 2A = 12.56 cm2
Calculate the volume:
V = r 2 x h
V = 12.56 x 6
V = 75.36 cm3
Sphere
Volume of a Cube
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)
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.
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:
Double Integrals as Volumes
dA= dy dx
dA= dx dy
Fubini’s Theorem for Calculating Double Integrals
Suppose that we wish to calculate the volume under the planeZ = 4 - x - y
over the rectangular region R: 0 ≤ x ≤ 2 , 0 ≤ y ≤ 1
in the xy-plane. then the volume is:
where A(x) is the cross-sectional area at x. For each value of x, we may calculate A(x) as the integral
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:
1
2
If we just wanted to write a formula for the volume, without carrying out any of the integrations, we could write
Fubini’s Theorem
If ƒ(x, y) is continuous throughout the rectangular region then:
R : a ≤ x ≤ b, c ≤ y ≤ d
Examples to Finding the volume using Double integral
Q#1
Solution:
Q#2
Solution:
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.
