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# Integrals

### Integration of polynomial quotient and partial fractions

A example is shown whereby the velocity of an object is expressed as a quotient of two polynomials in time t. Such that velocity v = (t^2 + 18t + 21)/{(2t+1)*(t+4)^2} The problem asks to determine the displacement of the object after time t = 2.1s. The process requires the integration of the polynomial between the limit

### Contour integral involving branch point singularity

By considering the integral of (z^2+1)^(-a) around a suitable contour C, prove: Integral from x=0 to x=infinity of dx/(x^2+1)^a = sin(pi*a) Integral from u=1 to u=infinity of du/(u^2-1)^a where 1/2 < a < 1. (Include proofs that the integrals over any large or small circular arcs tend to zero as their radii tend to in

### Volume of solids of revolution..

Volume of solids of revolution.. 1. A paraboloid dish (cross section ) is 8 units deep. It is filled with water up to a height of 4 units. How much water must be added to the dish to fill it completely? 4. Write an integral that represents the volume of the solid formed by rotating the region bounded by , , , and

### indefinite integral calculus

1. A quantity of gas with an initial volume of 1 cubic foot and a pressure of 500 pounds per square foot expands to a volume of 2 cubic feet. Find the work done by the gas. (Assume that the pressure is inversely proportional to the volume; that is, the gas obeys the ideal gas law such that PV = constant.) 2. Find the ind

### improper rational function

Integration by partial fractions 6. Solve for : 10. Evaluate: 11. Evaluate: 19. Express the improper rational function as the sum of a polynomial and a proper rational function:

### polar coordinates calculus

Use double integration in polar coordinates to find the volume of the solid that lies below the given surface and above the plane region R bounded by the given curve. 1. z=x^2+y^2; r=3 Evaluate the given integral by first converting to polar coordinates. 2. ∬_(0,x)^1,1▒〖x^2 dy dx〗 Solve by double i

### Taylor polynomial, Lower and Upper sums

2a. Find the second order Taylor polynomial for f(x) = x^(1/3) about x = x0, x0 > 0. 2b. Show that the function G(x) is differentiable at x0 and find G'(x0). 3a. Find an expression for the lower sum L(f,D) and upper sum U(f,D). 3b. Determine the lower integral and upper integral. Please refer to the attached images f

### Integration of trigonometric functions

I am supposed to find or evaluate the integral of cos^(2)XsinXdx I was thinking that i should use integration by parts but I am not sure what values to use for U, and dv...

### Maximum Overhang of a Pile of Bricks

See attached file.

### Trapezoidal Rule Word Problems

Please show ALL work! A bacteria population grows at a rate proportional to its size. Initially the population is 10,000 and after 5 days it's 30,000. What is the population after 10 days? How long will it take for the population to double? A solid S is generated by revolving the finite region bounded by the y-axis

### derivative and the slope of a curve

In this explanation, I am being asked to discuss the relationship between the slope of a secant line, the slope of a tangent line and the derivative AND in addition, I must explain the relationship between the area of a finite number of rectangles under a curve and an infinite number of rectangles under a curve and the definite

### Differential Equations and Indefinite Integrals

1. Find the solution of the differential equation dy/dx=x^3 -x, with the initial condition y(0) = -3. 2. Find the solution of the differential equation dy/dt=t^2 / (3y^2), with the initial condition y(0) = 8. 3. Find the solution of the differential equation dy/dx=(x=2)y^(1/2), with the initial condition y(0) = 1. 4. Fi

### Calculaion of work done on a spring

1) Find the arc length of the curve f(x)=x3/2 - 1 over the interval [0,1] 2) Find the arc length of the curve f(x)=ln(cos x) over the interval [0,pi/4] 3) Find the arc length of the curve f(x)=1/6 x3 + 1/2 x - 1 over the interval [1,2] The Next TWO Problems Refer To The Paragraph Below. This is a two step module. First you

### Definition of a Derivative

Calc Proofs 1) Using the following two functions X and X^2 develop their derivatives using the Definition of a Derivative for three values of "h" h = .1 h = .01 h= .001 and then repeat the calculation in the limit as h->0 2) Using the two functions above, show that the Finite Sum approximation of the area

### 13 review questions in calculus

The integrals wouldnt paste from word so I had to write them in 1. If an improper integral is found to have a finite solution, then: A) The solution will always be some multiple of &#960;. B) You've done something wrong. C) The function being integrated converges. D) The function being integrated diverges. 2. If f(x) =

### Integrating Technology in Math Classroom

Explain the importance of appropriately integrating Texas Instruments' CBL 2 (Calculator Based Laboratory), CBR 2 (Calculator Based Ranger), Fathom, a powerful data analysis software tool from Key Curriculum Press, and Excel into the middle school or high school mathematics classroom curriculum. Describe the benefits of their us

### Integrals and Derivatives

15. A) B) C) D) None of the above 16. What are the values of C0 and C1 in d(t) = C1 + C0t - 16t2, if d(1) = 4 and v(2) = -65? A) C0 = -1, C1 = 21 B) C0 = 1, C1 = -21 C) C0 = -1, C1 = 19 D) C0 = 0, C1 = 1 17. What does du equal in &#8747;2x(x2 + 1)5 dx? A) 2x B) 2u du C) 2x dx D) 5u4 18. What is

### Derivatives and Integrals

1. Let's say that f ''(k) = 0 @(13, -2). What does this mean? A) There is definitely an inflection point at that location. B) There might be an inflection point at that location. C) There definitely is not an inflection point at that location. D) There's no way to tell without first knowing what the specific function is.

### Integers and Rational Numbers

1)For any integer a, argue that a + 3 > a + 2 3) An integer a is divisible by an integer b means there is an integer z such that a = b x z. use any properties of the integers through page 14 to prove that fir integers a,b and c such that if a is divisible by b and b is divisible by c the a is divisible by c. 4) let Z deno

### Integration by Substitution using Chain Rule

WEEK 8 PARTICIPATION QUESTIONS... 1) Integration by substitution comes from Chain rule. Integration by parts is a consequence Product Rule of Derivatives. Prove that... [f(x) g(x)]' = f(x) g' (x) + f(x) g(x) Take the integral with respect to x of both sides of the equation, what will happen? 2) Retirement annuity.

### Multiple Integrals and vector calculus

Multiple Integrals and vector calculus. 1. To find the mass of a thin-shelled ellipsoid, you have to: A) Determine the inside surface area and then multiply by the shell's thickness B) Determine the average of the inside and outside surface areas and then multiply by the shell's thickness C) Find the difference

### Triple integral

Use a triple integral to find the volume of the given solids. 1.)the tetrahedron bounded by the coordinate planes and the plane 2x + 3y + 6z = 12 2.)the solid enclosed by the parabolas z = x^2 + y^2 and z= 0 and x + z = 1

### Brownian Motion

Please provide a detailed solution to the attached file. We are interested in estimating the response function of a Neuron...

### Evaluation of the definite integral

The problem is read, set up and evaluate the definite integral for the area of the surface generated by revolving the curve. y=(x^4/8)+(1/4x^2),1 is less than or equal to t and t is less than or equal to 3, about the x axis. also there's a part b which reads.x=1-t^2,y=2t, 0 is less than or equal to t and t is less than or equal

### Mathematics - Calculus - Integration in Commerce

Profit over the useful life of a machine - Suppose that when it is t years old, a particular industrial machine generates revenue at the rate R'(t) = 6,025 - 8t^2 dollars per year and that operating and servicing costs accumulate at the rate C'(t) = 4,681 + 13t^2 dollars per year. a) How many years pass before the pro

### Differential Equations Formatting

See attached for formatting 1. Considering the differential equation y' = (y/x)3 : a. Discuss existence and uniqueness of solutions. b. Determine if exist constant solutions. c. Determine the general integral. d. Solve Cauchy problems y(3) = -1, y(3) = 0 and determine the maximal interval of solutions. 2) Integrate the

### Line integrals

See attached page Evaluate the line integrals, where C is the given curve

### Integral

See attached page Evaluate the integral by making an appropriate change of variables.

### application of triple and double integrals

See the attached file for full description. 26. Evaluate the triple integral, where E is bounded by the planes y = 0, z = 0, x + y = 2 and the cylinder y^2 + z^2 =1 in the first octant. Find the volume of the given solid 30. Under the surface z = x^2y and above triangle in the xy-plane with vertices (1, 0), (2, 1), and (4

### Basic Mathematics Problems Involving Digits and Word Problems

Find the greatest common factor for the following group of numbers: 24, 36, 48 A rectangular picture window is 5ft by 8ft. Polly wants to put a trim molding around the window. How many feet of molding should she buy? (Can you draw a picture?) Write the following in standard form of a number (Hint: use a number char