Share
Explore BrainMass

Integrals

Integrals, Vector Fields, and Differential Equations

Please see attached 1) To find the general integral of the differential equation, discuss existence et uniqueness and find the particular integral that passes for the point (1;5/2) 2) Considering the linear equation of the 2nd order z'' - (2/x) z' + (2/ x2)z = 10 / x2 , x > 0 .

Integration and other topics

See attachment for fomatting 1 Evaluate 3∫1 1-∫-2 (x2y-2xy3)dydx 2 Correctly reverse the order of integration, then evaluate 1∫0 1∫y xeydxdy 3 The plane region R is bounded by the graphs of y=x and y=x2 . Find the volume over R and beneath the graph of f(x, y) = x + y. 4 Find t

An example of contour integration using Cauchy's formula

Evaluate the contour integral of z^2/(4-z^2) around the circle |z+1|=2. The question is attached in correct mathematical notation, along with the student's (incorrect) initial attempt. You will need to refer to this initial attempt when reading the solution.

Double Integrals

Calculate the double integral. See attached page for the problems.

Integration

1) Int. Sinx dx. = ( Pi/2 to 3/2pi) ie definite integral of sinx with lower limit pi/2 and upper limit 3/2 pi

Integrals of Trigonometric Functions

1) Evaluate. ∫ sinh^6 x cosh x dx 2) Evaluate. ∫ [ (√x+4)^3 / 3√x ] dx 3) Evaluate. ∫ x^2 sin(2x) dx 4) Evaluate. ∫ sin^5 x dx Please see attachment for actual sample problems.

Wave Equations and Periodic Differentiable Functions

3. Solve the wave equation, &#8706;2u/&#8706;t2 = c2(&#8706;2u/&#8706;x) -&#8734; < x < &#8734; With initial conditions, u(x,0) = (1/x2+1)sin(x), and &#8706;u/&#8706;t(x,0) = x/(x2+1) 4. Suppose that f is a 2&#1087;-periodic differentiable function with Fouier coefficients a0, an and bn. Consider the Fourier coeffici

Integrals, Area under the Curve and Solid of Revolution

1. Evaluate: &#8747;2cos2 xdx 2. Figure 12.1 y = 9-x2 , y=5-3x Sketch the region bounded by the graphs of Figure 12.1, and then find its area. 3. Figure 13.1 1?0x4dx Approximate the integral (Figure 13.1); n=6, by: a) first applying Simpsonfs Rule and b) then applying the trapezoidal rule. 4. Find

Area of a Region between Two Curves

LetR be the region bounded by the graph of f(x)=3x^2+6x and g(x)=18x-5x^2. 1) determine the area of R 2) Determine the volume of the solid of revolution formed when R is revolved about the line y=18. Please answer in detail.

Inverse Hyperbolic Integral

Evaluate the integral. (integral from 0 to 1) dt/sqrt[16t^2+1] Use Hyperbolic inverse and show steps please!

Integrals and differentiation

Differentiate the function f(x) = ln(2x + 3). Find . lim e^ 2 x/(x+5)^3 &#8594;&#8734; Apply l'Hopital's rule as many times as necessary, verifying your results after each application. Evaluate &#8747; x sinh(x)dx . Determine whether 2 &#8747; (x / ^(4-x^2)) (dx)

Integration

Evaluate &#8747;3x+3 / x^3-1 (dx) Use trigonometric substitution to evaluate &#8747;1 / ^/¯1+x2(dx) Determine whether converges or diverges. If it converges, evaluate the integral. &#8734;&#8747;-&#8734; 1 / 1+x2 (dx)

Integrals

Evaluate &#8747;(^/¯x+4)^3 / 3^/¯x(dx) &#8747;x2sin2x dx &#8747; sin5xdx

Integration

Find an upper and lower bound for the integral using the comparison properties of integrals. 1&#8747;0 1 /x+2(dx) Apply the Fundamental Theorem of Calculus to find the derivative of: h(x)= x&#8747;2 ^/¯u-1dx Evaluate: 4&#8747;1 (4+^/¯x)^2 / 2^/¯x (dx) Evaluate: &#8747;2cos^2 xdx Sketch

Area Bounded by a Smooth Simple Closed Curve

Show that if R is a region in the plane bounded by a piecewise smooth simple closed curve C then area is given by.... Please see the attached file for the fully formatted problem.

Integrals and Displacement of an Object

Evaluate a. b. c. d. An object is moving so that its velocity after t minutes is meters per minute. How far does the object travel from the end of minute 2 to the end of minute 3? a. 31 meters b. 61 meters c. -29 meters d. -9 meters

Integration using Spherical Bessel Function

Given a spherical Bessel function J1(w) = (1/w^2)*(Sin(w)-w*Cos(w)) Show that (pi)*x/2, -1<x<1 integral of J1(w)*Sin(wx)dw from 0 to infinity = 0, |x|>1 Need a through explanation!!

Composite Trapezoidal Rule, Simpson's Rule and Gaussian Quadratures

1. Use the composite Trapezoidal Rule with indicated values of n=4 to approximate the following integrals See Attached file for integrals. 2. Use the Excel programs for Simpson's composite rule to evaluate integrals in Problem 1. 3. Use Gaussian Quadratures with n = 2, n = 4, n = 5 to evaluate integrals in Problem 1.