The double integral of over the rectangle is if this limit exists evaluate double integral

FUNCTION SEVERAL VARIABLES .... 13 2 Application of function several variable ..... EVALUATE DOUBLE INTEGRAL We shall evaluate integral , where , then PROPORTIES:... APPLICATION OF DOUB

HO CHI MINH CITY UNIVERSITY OF

TECHNOLOGY FACULTY OF APPLIED SCIENCE

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REPORT CALCULUS 2 CLASS: CC07 – GROUP 6

INSTRUCTOR : PHAN THỊ KHÁNH VÂN

2151247

TABLE OF CONTENTS

Work Assignment 2

TABLE OF CONTENTS 3

I DOUBLE INTERGRALS 4

1) Theorems 4

2) Application of double integral 7

II DERIVATIVE 10

1) Theorems 10

2) Application of derivative 11

III FUNCTION SEVERAL VARIABLES 13

1) Theorems 13

2) Application of function several variable 20

DEFINITION

The double integral of over the rectangle is

if this limit exists

EVALUATE DOUBLE INTEGRAL

We shall evaluate integral , where , then

PROPORTIES:

MORE THEOREM:

APPLICATION OF DOUBLE INTERGRAL:

A Caculate the area of region R

Find the volume V of the solid S that is bounded by the elliptic paraboloid , the planes x=3 and y=3 , and the three coordinate planes

Solution

First notice the graph of the surface in Figure a and above the square region

However , we need the volume of the solid bounded by the elliptic paraboloid

, the planes x=3 and y=3 , and the three coordinate planes

(b) The solid S lies under the surface above the square region

Now let’s look at the graph of the surface in (b) We determine the volume V by evaluating the double integral over R2 :

=

B The mass and the center of mass

2 Find the mass and the center of mass of the lamina that occupies the region D and has the given density function ρ D is the triangular region with vertices (0,0), (2,1) and (0,3); ρ(x, y)

clc;

clear allall ;

fun1 = @(x,y) 1/6.*(x.*(x+y))

fun1 = @(x,y) 1/6.*(y.*(x+y))

PARTIAL DERIVATIVE

SECOND ORDER PARTIAL DERIVATIVE

The graph of f is a paraboloid z = 4 - x – 2y and the plane y=1 cut paraboloid by the parabol C = z = 2- x , y=1 The slope of the tangent line T with parabol C at (1,1,1) is f1 2 1 1 ’x

(1,1) = -2 Similarly, the plane x =1 cut paraboloid by the parabol C : z = 3 -2y , x = 1 2 2

The slope of the tangent line T with parabol C at (1,1,1) is f (1,1) = -4.2 2 ’y

b The geometry meaning of derivative in a specific direction :

FUNCTION OF SEVERAL VARIABLES THEOREMS

QUADRATIC SURFACES

APPLICATION FUNCTION OF SEVERAL

VARIABLES

A.The range R of the trajectory

1.The range R of the trajectory of the projectile fired with initial velocity from the barrel made with the horizontal line at an angle is determined by the formula

vo = input("Input the initial velocity");

phi = input("Input the angle");

g = 9.8

R = (v0^2 * sin(2*phi))/g

To find the hypotenuse by Pythagorean formula

To find the length of leg by Pythagorean formula

To find the length of leg by Pythagorean formula

clc

clear all

syms choice

disp("Pythagorean theorem for finding hypotenuse/leg")

Choice = input("Enter you choice hypotenuse(0)/leg(1), input 0

or 1: ")

for choice = 0

leg1 = input("input your first leg: ")

leg2 = input("input your second leg: ")

hypotenuse = sqrt(leg1^2+leg2^2)

for choice = 1

h = input("input your hypotenuse: ")

leg = input("input your given leg: ")

remainleg = sqrt(h^2-leg^2)

C uniform accelerated motion problem

3 Calculate displacement related to initial velocity, acceleration and time

We have displacement formula in uniform acceleration motion

With is the displacement

v0 = input("Input your inital velocity: ")

a = input("Input your acceleration: ")

t = input("Input your time: ")

s = v0*t + (a*t^2)/2