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# Integrals

### Revenue Function and Definite Integral of Revenue Function

Revenue at day D = (200 + 10D - 100P)*P D refers to the day, with Monday being 1, Tuesday 2, etc up to Friday with a value of 5. The assignment is as follows: 1. If you are charging \$1 per cup, what is your revenue for each of the five days? What is your total revenue for the week? 2. What is the indefinite integral

### Stokes Theorem, Curl and Positively Oriented Hemisphere

7) Use Stoke's Theorem to evaluate curl F*dS S is the hemisphere oriented in the direction of the positive x-axis. 8) Use Stokes Theorem to evaluate C is the boundary of the part of the plane 2x + y + 2z = 2 in the first octant. 9) Suppose that f(x,y,z)= , where g is a function of one variable such that g(2) = -

### Green's Theorem, Positively Oriented Curve, Ellipse

Evaluate the line integral by two methods: (a) directly and (b) using Green's Theorem. ∫c xdx + ydy. C consists of the line segments from (0,1) to (0,0)...and the parabola y = 1 -x^2.... Use Green's theorem to evaluate the line intgral along the positively oriented curve. ∫c sin y dx + x cos y dy

### Integration Techniques and Applications

1. Find the indefinite integrals for the following functions: a. f(X) = 10000 b. f(X) = 20X c. f(X) = 1- X2 d. f(X) = 5X + X-1 e. f(X) = 12- 2X f. f(X) = X3 + X4 g. f(X) = 200X - X2 + X100 2. Find the definite integral for the following functions: a. f(X) = 67 over the interval [0,1] b.

### It is an explanation for finding the integral by the method of summation i.e., by evaluating the integral as limit of a sum(part 13). Evaluate the definite integral &#8747; sinh x dx, where the lower limit is a and the upper limit is b, i.e., integral of sinh x, where the lower limit is a and the upper limit is b as limit of a sum.

Calculus Integral Calculus(XIII) Definite Integral as the Limit of a sum Method of Summation

### Definite Integral as the Limit of a sum BrainMass Expert explains

Calculus Integral Calculus(I) Definite Integral as the Limit of a sum Method of Summation Definite Integral It is an explanation for finding the integral by using the method of summation(part I). Find by the method of summation the value of: (a) ∫ e^( - x)dx, where the lower limit is a and the upper limit is b

### Compute the following integrals where m and n are non-negative integers.

1. Compute the following integrals where m and n are non-negative integers. Look out for special cases. (a) &#8747; 0 --> L cos(n pi x/L)cos(m pi x/L) dx (b) &#8747; 0 --> L cos(n pi x/L)sin(m pi x/L) dx Please see the attached file for the fully formatted problems.

### Heat Equation with Circular Symmetry : Total Heat Energy, Flow of Heat Energy and Equilibrium Temperature Distribution

8. Heat Equation with Circular Symmetry. Assume that the temperature is circularly symmetric: u u(r,t), where r^2 x^2 | y^2. Consider any circular annulus a ≤ r ≤ b. a) Show that the total heat energy is r π f^b_a cpurdr. b) Show that the flow of heat energy per unit time out of the annulus at r b is: (see attachment

### Congruences, Primitive Roots, Indices and Table of Indices

6. Let g be a primitive root of m. An index of a number a to the base (written ing a) is a number + such that g+&#8801;a(mod m). Given that g is a primitive root modulo m, prove the following... 7. Construct a table of indices of all integers from.... 8. Solve the congruence 9x&#8801;11(mod 17) using the table in 7. 9.

### Integration Applications : Equation of Solution, Marginal Cost and Maximum Height

1.) You are shown a family of graphs each of which is a general solution of the given differential equation. Find the equation of the particular solution that passes though the indicated point. dy/dx=-5x-2 point (0,2) 2.) Find the cost function for the marginal cost and fixed cost marginal costs fixe

### Helicoid Integral

Evaluate &#8747;&#8747;S &#8730;(1 + x^2 + y^2) dS where S is the helicoid: r(u,v) = u cos(v)i + u sin(v) j + vk , with 0 &#8804; u &#8804; 3, 0 &#8804; v &#8804; 2pi. Please see the attached file for the fully formatted problem.

### Flux Integrals

Suppose is a radial force field,... is a sphere of radius...centered at the origin, and the flux integral.... Let be a sphere of radius... centered at the origin, and consider the flux integral... . (A) If the magnitude of... is inversely proportional to the square of the distance from the origin,what is the value of..

### Newton's Law of Cooling : Average Temperature

If a cup of coffee has temperature 95 degrees Celsius in a room where the temperature is 20 degrees Celsius, then according to Newton's Law of Cooling, the temperature of the coffee after t minutes is T(t)=20+75e^(-t/50). What is the average temperature of the coffee during the first half hour?

### Question About Average Value of a Function on an Interval

Find the numbers b such that the average value of f(x)=2+6x-3x^2 on the interval [0,b] is equal to 3.

### Lebesgue Integral and Monotone Convergence Theorem

Let f be a nonnegative integrable function. Show that the function F defined by F(x)= Integral[from -inf to x of f] is continuous by using the Monotone Convergence Theorem. From Royden's Real Analysis Text, chapter 4. See the attached file.

### Solve Sequences Used in Given Situation

There is a rope that stretches from the top of Maidwell building to a tree on the racecourse, and the length of this rope is 1km. A worm begins to travel along the rope at the rate of 1cm each second in an attempt to get to the other end. then a strange thing happens... some malevolent deity intervenes to make life even hard

### Evaluating integrals

Evaluate the following integrals: (1)The integral of (6sin[2x])/sin(x)dx=____+C (2)The integral of (7-x)(3+[x^2])dx=____+C (3)The integral from 3 to 7 of ([t^6]-[t^2])/(t^4)dt=____+C (4)The integral of (6sin[x])/(1-sin^2[x])dx=____+C

### Definite and indefinite integrals

Thank you in advance for your help. Evaluate the following definite and indefinite integrals: (1)The integral of (2x)/([x^2]+1)dx (2)The integral of [(arctan(x))/([x^2]+1)]dx (3)The integral of sqrt([x^3]+[1x^5])dx (4)The integral of (2+x)/([x^2]+1)dx (5)The integral of (2x)/([x^4]+1)dx (6)The integral from 0 to 2 of (x

### Proving an Integral Equation

If f is continuous for all real numbers, prove that the integral from a to b of f(x+c)dx=the integral from a+c to b+c of f(x)dx. For the case where f(x) is greater than or equal to 0, draw a diagram to interpret this equation geometrically as an equality of areas.

### Integrals : Rate of Change Word Problem

Water flows from the bottom of a storage tank at a rate of r(t)=200-4t liters per minute, where 0 is less than or equal to t and t is less than or equal to 50. Find the amount of water that flows from the tank during the first 10 minutes.

### Work Done by Radial Vector Field Along a Curve

If C is the curve given by r(t) = (1 + 3 sin t) i + (1 + 5 sin^2 t) j + (1 + 5 sin^3 t) k, 0 ≤ t ≤ π/2 and F is the radial vector field F(x, y, z) = xi + yj + zk, compute the work done by F on a particle moving along C.

### Integral of a Contour

Calculate the following integrals: &#8747; from 0 to &#8734; x^¼/(x²+9) dx Please see attached for proper format.

### Residues and Closed Contours : Solve the Integral

Calculate the following integral... Please see attached for full question. Solution. Consider a close contour C shown above, where C consists of and a line segment from -R and R. Consider positive orientation, namely, clockwise. Choose r large enough so that are in the region covered by C. Let . By residual Theorem

### Evaluating an integral in terms of areas

"Evaluate the integral by interpreting it in terms of areas: the integral as 0 goes to 8 of |5x-10|dx"

### Integrals : Riemann Sum with Diagrams

This question has me going around in circles. I can't make the Sigma symbol on the computer, so I used the word "Sigma" instead. For (c), n is above the Sigma symbol and i=1 is below it. (a)Find an approximation to the integral as 0 goes to 4 of (x^2-3x)dx using a Riemann sum with right endpoints and n=8. (b)Draw a diagram

### Elementary Numerical Analysis

GAUSSIAN NUMERICAL INTEGRATION 1. Consider approximating integrals of the form... in which f(x) has several continuous derivatives on [0, 1] a. Find a formula... which is exact if f(x) is any linear polynomial. b. To find a formula... which is exact for all polynomial of degree &#8804; 3, set up a system of four e

### Elementary Numerical Analysis : Gaussian Numerical Integration

1. Show that if an integration formula of the form In ( f ) = &#8721; wjf(xj) is exact when integrating 1, x, x2, ..., xm, then it is exact for all polynomials of degree &#8804; m. Please see attached for proper format of question.

### Line Integral of Curve which is the Union of Two Line Segments

Let C be the curve which is the union of two line segments, the first going from (0, 0) to (3, -1) and the second going from (3, -1) to (6, 0). Compute the line integral. Please see attached.

### Arc Length of Curve, Tangent Line, Limits and Solid Revolution

1. Find the volume of the solid generated by revolving the region enclosed by: {see attachment} 2. Find the arc length of the graph of the curve {see attachment} 3 - 7. Integrate attached equations ... 8. Find the limit of the improper integral: {see attachment} 9. Find the arc length of the curve given in parametric

### Indented Path : Integral of Branch of Multiple-Valued Function

Show that... by integrating an appropriate branch of the multiple-valued function... over (a) the indented path in Fig. 97, Sec 75; (b) the closed contour in Fig. 99, Sec. 77. See attachment for equations and diagrams.