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Centroid-triple integrals

Find the centroid of the first octant region that is interior to the to the two cylinders x^2+z^2=1 and Y^2+Z^2=1 centroid for x y and z are x'=1/M*triple integral of x^2*dV y'=1/M*triple integral of y^2*dV z'=1/M*triple integral of z^2*dV

Monte Carlo integration

Describe how to use the Monte Carlo method to estimate the double integral of xydxdy over the area 0<x<y and 2<y<4

Mathematical Methods

(a) Describe how the weights for the order 4 closed Newton-Cotes quadrature formula could be found. Do NOT calculate the weights. (b) What are composite quadrature rules and why are they preferred to using higher order quadrature rules? (c) What are the main characteristics of a predictor-corrector method for solving an initia

Line Integral : Green's Theorem

A) For what simple closed (positively oriented) curve C in the plane does the line integral of (e^(-x)+ 4x^2y +y)dx + (x^3-x*y^2+5x)dy have the largest positive value? (use Green's theorem) b) Determine what this value is.

Power Series

Problem: Given the power series for the following function (1+x)^k (a) Write the power series for (1+x)^(1/3) (b) Use the power series from part (a) to find the power series for x^3 (c) Using this series approximate the following integral (1+x^3) ^(1/3) using the first three terms

Solve the ODE (Integrate)

Find the explicit solution to the ODE 2yy'=(1+y^2) subject to y(0)=4. What is the solution if y(0)=-4? *(Please see attachment for proper citation of symbols and numbers)

Changing variables in multiple integrals

Using the coordinate change u=xy, v=y/x, set up an iterated integral for the polar moment of inertia of the region bounded by the hyperbola xy=1 , the x-axis, and the two lines x=1 and x=2. Choose the order of integration which make the limits simplest THIS MESSAGE IS ADDRESSED TO ANY TA: I found something , I just want you

Functions and Integrals

Please assist me with the attached problems relating to functions and integrals - thank you!

Limits of iterated integrals (parallel axis theorem)

Prim is primitive! In genral the moment of inertia around an axis( a line) L is: Isubl=double prim (dist(.,L)^2*delta*dA) The collection of lines parallel to the y axis have the form x=a .Let I=Isub(y) be the usual moment of inertia around the y axis I= double prim of x^2*delta*dA Let I(bar) be the moment of ine

Limits in iterated integrals

Compute the mass, centroid, and moments Ix, Iy and Io of the half-disk: y>0, x^2+y^2<1 with density delta(x,y)=y it is said we should know the primitive (sin(x))^n or (cos(x))^n from Pi/2 to 0

Double integration over the domain

Find the integral of f(x,y)=x^2 over the domain D which is bounded by y=3x, x=3y and x+y=4 Hint: use the transformation x=3u+v and y=u+3v

Contour integral

Find the integral over C of f(x,y) = x^2 where C is the unit circle


Evaluate integate (3sin2x - 2cos3x)dx a=pi/4 and b=pi/2

Definite Double Integral

Evaluate the attached integral: a) Write an equivalent iterated integral with the order of integration reversed. b)Evaluate this new integral and check that your answer agrees with part (a)

Double Integrals

Please see the attached file for full problem description. --- Find the volume of the region that lies under the graph of the paraboloid z = x^2 + y^2 + 2 and over the rectangle R = {(x, y) | -1 and in two ways (a) by using Cavalieri's principle to write the volume as an iterated integral that results from slicing

Integration: The Limit of a Sum

Find by the method of summation the value of : a) The integral (from 0 to 1) of the square root of x. (dx) b) The integral (from 1 to 4) of 1 divided by the square root of x. (dx) Please view the attachment for proper formatting.

Cauchy's Formula

See Attachment for equation We know that sin z and cos z are analytic functions of z in the whole z-plane, what can we conclude about *(see attachment for equations)* in the first quadrant

Integration as the limit of a sum

Find the integral by the method of summation the values of :- (a) integral of e^(-x) where the range of integration is from a to b. (b) integral of e^(kx) where the range of integration is from a to b.


Please help with various Calculus questions. (please see attachment)

Improper Integrals surface area problem..

Use l'hopital's rule if needed. Show all work step by step. Use proper notation please. Surface Area: The region bounded by [(x-2)^2]+y^2=1 is revolved about the y-axis to form a torus find the surface area of the torus. PLEASE show or explain step by step process.

Work in joules

A trough is 2 meters long, 2 meters wide, and 2 meters deep. The vertical cross-section of the trough... (See attached)