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Integrals

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

Evaluate the integral. (integral sign) e^(2x)dx/sqrt[1-e^(4x)] Please show steps.

R is bounded below by the x-axis and above by the curve

1)Figure 11.1: 0<=x<=(pie)/2 R is bounded below by the x-axis and above by the curve y = 2cos(x), Figure 11.1. Find the volume of the solid generated by revolving R around the y-axis by the method of cylindrical shells. 2)Figure 15.1: y= 1/(x^2+4x+5) R is the region that lies between the curve (Figure 15.1) and t

convergence of improper integral

See attachment and show work. 1. Find a function f(x) = x^k and a function g such that f(g(x)) = h(x) = (3x + x^2)^0.5; 2. Express the distance between the point (3, 0) and point P(x, y) of the parabola y = x^2 as a function of x 3. Determine whether converges or diverges. If it converges, evaluate the integral

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)

Implicit differentiation, integral, maximum area

Use implicit differentiation to find an equation of the line tangent to the curve x3 + 2xy + y3 = 13 at the point (1, 2). What is the maximum possible area of a rectangle inscribed in the ellipse x2 + 4y2 = 4 with the sides of the rectangle parallel to the coordinate axes? 3 . Evaluate &#8747; dt / (t+1)^

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.

Area under a curve description

• Find an estimate of the area under the graph of between and above the -axis. Use four left endpoint rectangles. • Find an estimate of the area under the graph of between and above the -axis. Use four right endpoint rectangles. • Find an estimate of the area under the graph of between and . Use four left

Proofs : Riemann Integrable Functions

Let RI be the set of functions that are Riemann Integrable. Disprove with a counterexample or prove the following true. (a) f in RI implies |f| in RI (b) |f| in RI implies f in RI (c) f in RI and 0 < c <= |f(x)| forall x implies 1/f in RI (d) f in RI implies f^2 in RI (e) f^2 in RI implies f in RI (f) f^

Integration: Calculation of Distance Travelled

1. In a mechanical system, the displacement of a body, s meters, is related to time t seconds, by the integral: t = (integral) [r/(g +ks)]ds where r, g and k are all constants. Determine an expression in terms of r and k, for the time taken by the body to travel a distance g/k meters from its initial position.

Integral Domain, Localization and Maximal Ideals

Please see the attached file for the fully formatted problems. Let A be an integral domain. For a prime ideal P C A and let S = A P be the complement of P in A. Observe that S is multiplicatively closed (WHY?) and form the subring A_p := S^-1 A of the field of fractions F of A. We regard A as a subring of A_p as usual - i.e.