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

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 2 A = 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 plane Z = 4 - x - y over the rectangular region R: 0 ≤ x ≤ 2 , 0 ≤ y ≤ 1 in the xy-plane. then the volume is:

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:

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.