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# Integrals

### Integrals : Finding Work Done

A 100 ft length of steel chain weighing 15 lb/ft is hanging from the top of a tall building. How much work is done in pulling all of the chain to the top of the building? keywords: integration, integrates, integrals, integrating, double, triple, multiple

### Finding volume of a solid of revolution

Find the volume of a solid that is generated by rotating the region formed by the graphs of y=x^2, y= 2, and x = 0 about the y-axis?

### Graph : Finding the Area of a Shaded Region

(See attached file for full problem description with image) The graph below represents the function f(x) = x3 + 2x2 - 5x - 6. Explain how you process the calculation of the shaded region.

### Integration of exponential

Integration of exponential: I am having difficulty integrating exponential. I want to compute the indefinite integral of r`(t) where r`(t) is the vector <3, -e(^ -t), 0> The first integration I have is <3t+c1, ------- c3> I am not sure how to integrate the -e^-t ------------------------------------------

### FInd the area and avearge value by integration

1. Determine the value of f(x)=-5x^3+7x on [-1,2]. 2. Determine the area of region bound by f(x)=-2x^2+x+6 and the x-axis on [0,3]. 3. Determine the area of the region bounded by f(x)=x^2 - 4 and g(x)=x+2 4. Determine the area of the region bounded by f(x)=x+6 and g(x)=-x^2+2 on the interval on [-2, 2]. keywords:

### Converting a double integral into a triple integral.

I have a double integral of the form I from 0 to 2 and I from 0 to y (4-y^2) dx dy How does one convert this double integral into a triple integral? keywords: integration, integrates, integrals, integrating, double, triple, multiple

### Finding area under normal curve

Find the area under the normal curve which is shaded on the graph. Use 4 decimal places. See attached file for full problem description.

### Real Analysis : Fredholm equation and Lipschitz condition

Consider the nonlinear Fredholm equation where is continuous on [a,b] and is continuous and satisfies a Lipschitz condition: on the set . Show that the integral equation has a unique solution on [a,b] if .

### Proof involving integral

(See attached file for full problem description with proper symbols) --- Assume that f is continuous on [a,b], g is differentiable on [c,d], g([c,d]) [a,b] and F(x) = For each x [c,d]. Prove that F'(x)=f(g(x))g'(x) For each x (c,d). ---

### Integral calculus to solve differential equation problems

1. The United States Census Bureau mid-year data for the population of the world in the year 2000 was 6.079 billion. Three years later, in 2003, it was 6.302 billion. Answer the following questions. (See attached bmp file) 2. A metal ball, initially at a temperature of 90 C, is immersed on a large body of water at a temperat

### Integral Creation

Create an integral whereby you are forced to use all four types of integration. Work the problem and explain why each (u-substitution, trig substitution, fractions, parts) are all needed. this must be only one integral, that is it must all be under a singular fraction and cannot be the sum such as integral of lnx+arctanx dx or

### Evaluation of improper integrals using residues theorem

Please provide in-depth evaluation of improper integrals using residues theorem.

### Delta Functions: Example Problems

(See attached file for full problem description) 1. Calculate the values of the following integrals. 2. Write delta(x^2 - 4) in terms of a sum of ordinary delta functions. 3. Write delta(sin(x)) in terms of a sum of ordinary delta functions (an infinite number!). 4. Calculate the values of the following integrals inv

### Integration and Simpson's Rule

(See attached file for full problem description with proper symbols and equations) --- A. Evaluate the improper integral: Infinity ∫ (xe^x^2)dx 0 B. Complete the square, then use integration tables to evaluate the indefinite integral: ∫ {(sqrt(x^2 + 6x + 13))/x+3}dx C. Which of the following would

### Evaluating an Integral: Example Problem

pi/2 ∫ (sin^2 theta x cos theta + 2 sin^4 theta x cos theta) dtheta 0 pi/2 = [sin^3 theta / 3 + 2 sin^5 theta / 5] 0 As you can see, all of the cosines are gone. I suspect that there is

### Triple Integrals : Volume of a Solid in Spherical Coordinates

Consider the solid inside the surface X^2 + Y^2 +Z^2 = 9 and outside the surface X^2 +Y^2 +Z^2 = 1 a) Use SPHERICAL coordinates to to write the integral to calculate the volume of the solid. b) Calculate the integral from part a keywords: integration, integrates, integrals, integrating, double, triple, multiple

### Volume of a Solid using Rectangular and Polar Coordinates

Consider the solid bounded above by the plane Z = 4 and below by the circle X^2 + Y^2 = 16 in the XY-plane. a) Write the double integral in rectangular coordinates to calculate the volume of the solid. b) Write the double integral in polar coordinates to calculate the volume of the solid. c) Evaluate part a or part b

### Integration: volume of solid

Let f and g be the functions given by f(x) = 1 + sin(2x) and g(x) = e^(x/2). Let R be the shaded region in the first quadrant enclosed by the graphs of f and g. The region R is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are semicircles with diameters extending from y=f(x) to y=g(x).

### Convergence Tests

Using one of the tests for convergence (ratio, root, comparison, limit, integral, nth term, etc.), show whether the following series converges or diverges: &#8734; &#8721; (3^n) / n³(2^n) n=1

### Triple Integral Example Problem

Please provide a detailed, step by step solution for the following problem using a triple integral if possible. Find the volume inside the paraboloid Z = X^2 + Y^2 below the plane Z = 4 Thank you.

### Applications of integration

Let R be the shaded region bounded by the graphs of y=sqaure root of x, and y=e to the power of -3x, and the vertical line x=1. a) Find the area R b) Find the volume of the solid generated when R is revolved about the horizontal line y=1. c) The region R is the base of a solid. For this solid, each cross section perp

### Volume of a Solid of Revolution by Shell Method

Approximate the volume of the solid generated by revolving region formed by the curve y=x^2, x-axis and the line x=2. Volume approximated by concentric shells a) Sketch the reqion y=x^2, x-axis and the line x=2. b) We'll approximate the volume revolving the region about the y-axis. c) partition the interval [0, 2) in x,

### Indefinite integral using trigonometric substitution : &#8747;x^2/&#8730;(25-x^2)

&#8747;x^2/&#8730;(25-x^2) To solve a given indefinite integral using a suitable trigonometric function substituted. Please see the attachment for the problem.

### Find the centroid of a lamina using Definite integrals when density of the region is a variable

A lamina is described with the help of three curves, given that density of the lamina at a point P is inversely proportional to its distance from y-axis and density at one of the points in the region of the lamina is also given. We need to find the centroid of the region. For full text of the problem, please find the attachm

### Find the centroid of a lamina using Definite integrals when density of the region is a variable

A lamina is described with the help of three curves, given that density of the lamina at a point P is proportional to the square of its distance from x-axis and density at one of the points in the region of the lamina is also given. We need to find the centroid of the region. For full text of the problem, please find the at

### Quantitative Methods Questions

I have completed the answers to the questions. I just need to have someone confirm that they are correct. Thank you! True/False Indicate whether the sentence or statement is true or false. F 1. Management science is the application of a scientific approach to solving management problems in order to h

### PID (Proportional / Integral / Derivative) Control System

Consider the system shown in Fig.1. (attached file) This is a PID control of a second-order plant G(s). Assume that disturbances D(s) enter the system as shown in the diagram. It is assumed that the reference input R(s) is normally held constant, and the response characteristics to disturbances are a very important consideration

### Integration / Anti-derivative (8 Problems)

Please show how to solve each of the following problems. Find the antiderivative (integral) 6. (x^2/the square root of [x^3-4])dx 8. (x^2 - 2)^3 2x dx 9. sin^3(x)dx 10. x^3/x^2 + 1 dx 11. 1/xln x dx 12. ln x/x dx 14. 2x + 1/square root of [x + 4] dx

### Verifying the Formulas for Volumes of a Cone and a Sphere

Please explain how to solve the following: 1. Use the disk method to verify that the volume of a right circular cone is 1/3pir^2h. 2. Use the disk method to verify that the volume of a sphere is 4/3pir^3.

### Integration : Finding Area of a Triangle Given its Vertices

Use integration to find the area of the triangle having the given vertices. (0,0) (a,0) (b,c)