Let f(x) be a positive continuous function on [0,1/2], f(x) =< 1/2. Let A = { (x,y) : 0 =< x = 1/2, 0=<y=< f(x)} Prove that; m*(A) = integral from 0 to 1/2 of f(x)dx. Please I don't want a solution from a book, I want a simple proof based on basic definitionsand given info.
Find the volume of y = 1/sqrt(1+x^2) bounded by y=0, x=-1, x=1 I'm using the disc method with a dx function: V = pi integral( [R(x)]^2 ) dx Therefore, I have V = pi integral( [1/sqrt(1+x^2)] ^2 ) dx from -1 to 1 = pi integral( [ 1/(1+x^2) ] ) dx from -1 to 1 I can't figure out how to integrate. Please explai
I need to find the arc length of y = 1/6 x^3 + 1/(2x) on the interval [1,3]
Find the arc length of the graph of the function over the indicated interval: y=1/6 x^3 + 1/(2x^2), [1,3] I know S = Intergral( sqr( 1 + [f'(x)]^2 )) dx from 1 to 3. I get y' = [ 1/9 x^4 - 1/3 + 1/(4x^4) ] dx Therefore, S = Intergal( sqr( 1 + 1/9 x^4 - 1/3 + 1/(4x^4) )) dx from 1 to 3 = Intergal( sqr( 2/3 + 1
Let be a positive function in . Define a new function by Prove that . Please see the attached file for the fully formatted problems.
Find the integral of: ((1+(sinh t)^2)^(1/2))dt.
Let X be a normed space, I closed interval ( or half-open on the right) and a = inf I, b = sup I. Let h : I -> [0,infinity) be a continuous function such that integral ( from a to b ) h(t)dt < positive infinity where integral from a to b represents the improper integral when I is not closed. Let epsilon > 0 and
If is a measure space and , show that defines a bounded integral operator. Please see the attached file for the fully formatted problems.
Please see the attached file for the fully formatted problems.
Use shells to find the volume of the solid formed by revolving the given region about the y-axis. 22) the region bounded by the curve y=SQRT(x), the y-axis, and the line y=1. 24) the region bounded by the parabolas y=x^2, y=1-x^2, and y axis for x≥0. 26) the region inside the ellipse 2(x-3)^2 + 3(y-2)^2 = 6 about
Find the volume of the solid formed when the region described is revolved about the x axis using washers and disks. 14) the region under the curve y= cubed root of x on the interval 0≤x≤8. 16) the region bounded by the lines x=0, x=1, y=x+1, and y=x+2. 20) the region bounded by the curves y=e^x and y=e^-x on
Which is the acceptable trial solution? (See attached file for full problem description)
Show that the solution of: dM/dt = Poert - pM M(0)=0 is M(t) = [Po/(r + p)](ert - e-pt) Please see the attached file for the fully formatted problems.
1. Use integration to find a general solution of the differential equation. dy / dx = (x-2)/ x = 1 - 2/x 2.Solve the differential equation. dy / dx = x + 2
Let H be the collection of all absolutely continuous functions f [0,1] -> F, where F denotes either real or complex field ) such that f (0) = 0 and . If for f andg in H, then H is a Hilbert space. Please see the attached file for the fully formatted problem.
Please see the attached file for the fully formatted problem.
Please see attached
Hello. Thank you for taking the time to help me. I cannot use mathematical symbols, thus, * will denote a partial derivative. For example, u*x denotes the partial derivative of u with respect to x. To simplify things, I will let p=u*x and q=u*y. Furthermore, I will use ^ to denote a power. For example, x^2 means x squared. Also,
Suppose that is analytic on and outside the simple closed negatively oriented contour . Assume that is analytic at and . Prove by formulating Cauchy's integral formula for in an annulus and letting the outer radius tend to that for all outside .
Hello. Thank you for taking the time for helping me. The following is the problem which I need to solve (there are actually two parts): I need to construct Green's function for the Dirichlet problem (Laplace's equation) in the upper half plane R={(x,y) : y>0} and I must derive Poisson's integral formula for the half plane.
Using the method of residues, verify the following: (See attached file for full problem description).
(See attached file for full problem description and embedded formulae) --- Why can (1) be regarded as a special case of (2)? (1) Cauchy's Integral formula (no need to prove): is a simple closed positively oriented contour. If is analytic in some simply connected domain D containing and if is any point inside ,
1) a. If g is measurable on [a, b] with g(x) 0 for all x [a, b], prove that 1/g is measurable on [a, b]. b. prove that every continuous real- valued function f on [a, b] is measurable. c. if f is differentiable on [a, b], prove that f' is measurable on [a, b]
7.. Given the vector field F(x,y,z) = xi + (x+2y+3z) j + z2 k Let C he the circle on the xy-plane, centered at the origin (0,0) and having as radius r=5. Let S be the part of the paraboloid z = 16? x2 ? y2 which lies above the xy-plane (z ≤ 0). Use the Stokes's Theorem to evaluate the line integral of this vector field a
Let S be the portion of the circular cone in a space that has an equation z^2= x^2 + y^2 and that lies between the planes z=7 and z=11. Given the scalar function...evaluate the surface integral over... (See attached file for full problem description)
Prove the following: Let be a sequence of functions continuous on a set containing the contour , and suppose that converges uniformly to on . Then the series converges to . Using this result and the Generalized Cauchy Integral formula for derivatives (see below), show the following: If all are analytic
Without evaluating the improper integrals and find the numerical value q of their quotient by considering the loop integral where is the semi-circular loop indented at the origin. Explain why Jordan's Lemma (see below) is inadequate here, and write a complete formulation of a more general Jordan's lemma
Please see the attached file for the fully formatted problems.
Please see the attached file for the fully formatted problems.
Use residues to evaluate integral of a trigonometric function. Please see the attached file for the fully formatted problems.