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Integrals

Real analysis - Step Functions and Riemann Integrals

If f is defined on [a,b] and and are, respectively, a nondecreasing and a nonincreasing sequence of step functions such that for all k and all and for almost all , show that f is Riemann integrable on [a,b]. Notes from section of book below: Section 5 Notes: Theorem 5.1 If f is Riemann integrable on [a,b],

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.

Integrals, Rate of Change and Recangle Inscribed in an Ellipse

(See attached file for full problem description) I need help with 7C, 10 and 12 7. Find the integral... 10. A beacon on a lighthouse 2000 m away from the nearest point P on a straight shoreline revolves at the rate of 10 pi radians per minute. How fast is the beam of the light moving along the shoreline when it is 500 m fro

Real Analysis : Integral Proof

If X(c,d) is the function from into such that X(c,d)(x)={1 if xE(c,d) and (c,d)⊂[a,b], show that ∫a-->b Xc,d =d-c. {0 otherwise Please see the attached file for the fully formatted problem.

Proof : Show integral is zero.

If Xc is the function from to such that Xc(x)={1 if x=c and cE[a,b] , show that ∫a-->b Xc =0. {0 if x≠c Please see the attached file for the fully formatted problems.

Proof using Riemann Integrals

If and are functions from to which are Riemann integrable on and which differ at only a finite number of points in , show that . Please see the attached file for the fully formatted problems.

Integrals

4 A) Evaluate ∫ 6x(2x^2 - 1) dx 2 b) Write down a definite integral that will give the value of the area under the curve y=x^3 cos (1/4 x) between x=pi and x=2pi pi=3.142 (you are not asked to evaluate the integral by hand) c) use mathcad to find the area described in par

G(x)= 19+15^3/x

Find the indefinite integral of the following function. g(x)= 19+15^3/x

Proof involving integral

(See attached file for full problem description) Assume that f is continuous on Reals and periodic with p. Show that for any a

Indefinate integrals-

Find the indefinate integrals off the following functions: f(t) = 2cos(4t) - 3e^5t g(x) = (19+15x^3)/ x (x>0) h(u)= sin^2[(1/10)u] I'm struggling to understand these questions so if someone could write a/n solution/explanation for them that would be great.

Proof with integral

(See attached file for full problem description) --- Assume that f is continuous on [a,b] and f(x) 0 for each x [a,b]. Prove that >0 if there exists c (a,b) such that f(c)>0.

Calculus: Solve differential equation problems by using integral calculus

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

Evaluate Calculus Problems

(See attached file for full problem description with proper equations) Evaluate these: 1. Find dw/dx for the function given by w=xye^(xyz) 2. Find the center and radius of the sphere given by the equation x^2 + y^2 + z^2 + 4x - 2y + 2z=10 3. The sum of $2000 is deposited in a savings account earning r percent int

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

Solve the improper integral

(See attached file for full problem description) I tried solve this problem by following Cauchy's Residue Theorem. However, the answer is always wrong.

Application of Residues

I want to find the Cauchy value by using residues. (See attached file for full problem description)

Evaluate the Integrals

A. Evaluate ∫ x(sqrt(x+1))dx B. Find the area bounded by y= x/(1+x)^2, y=0, x=0, and x=2 C. Evaluate ∫ 1/x(sqrt(x+9))dx D. Find the indefinite integral using integration by parts: ∫x^2(e^2x)dx Infinity Evaluate the improper integral: ∫ ln(x)dx

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 followin

Evaluate integrals

(See attached file for full problem description with proper questions) 1. Find the indefinite integral 2. Find the definite integral:(4x+1)1/2 dx 3. Find the area of region bound by the graphs of the equations, then use a graphing utility to graph the region and verify your answer: Y=x(x-2)^(1/3) Y=0,

Triple integral problem 9B

Let Q be the sphere: X^2 + Y^2 + Z^2 = a^2 a) Use CYLINDRICAL coordinates to set up the integral to calculate the volume of Q b) Use SPHERICAL coordinates to set up the integral to calculate the volume of Q c) Solve for Q using either a or b

Double Integral : Volume of a solid - polar and rectangular 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: ∞ ∑ n(2^n)(n + 1)! / (3^n)n! n=1

Explanation to Convergence Tests

Using one of the tests for convergence (comparison, limit, integral, nth term, etc.), show whether the following series converges or diverges: infinity E (1 + cos n)/ n^2 n=1