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Integrals

Continuous Functions, Fundamental Set of Solutions

Consider the attached differential equation where I = (a,b) and p,q are continuous functions on I. (a) Prove that if y1 and y2 both have a maximum at the same point in I, then they can not be a fundamental set of solutions for the attached equation. (b) Let I = {see attachment}. Is {cos t, cos 2t} a fundamental set of solu

Integral

See attached... Let g(a) be the real solution to x+x^5=a

Area Between a Curve

Please assist me with the attached problems relating to finding the region within a curve. 3. (a) Obtain an expretsian far Calculating the area between the curve y=2?x+x2 and the u-axis far 0 <x< 2 by dividing the area up into 2n strips of equal width (each strip will have width 1/n) and then taking the limit as n ---> infini

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

Integration

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