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Integrals

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.

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

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

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^

Integrals : Average Value of a Function

Please see attached file for full problem description. 1. What is the average value of the function f in Figure 6.4 over the interval ? From the graph, we can approximate: The average value of f on the interval from 1 to 6 is 3. Find the average value of over the interval [0, 2]. The average value

Definite Integrals and Rate of Change

Please see attached file for full problem description. 16. An old rowboat has sprung a leak. Water is flowing into the boat at a rate given in the following table. t minutes 0 5 10 15 r(t), liters/min 12 20 24 16 (a) Compute upper and lower estimates for the volume of water that has flowed into the boat during the 1

Estimates of Velocity, Distance and Area under the Curve

Please see attached file for full problem description. Section 5.1 p. 224: 7 7. Figure 5.4 shows the velocity, v, of an object (in meters/sec). Estimate the total distance the object traveled between t = 0 and t = 6. We can estimate this using 1 second intervals. Since the velocity is increasing on the interval

Velocity, Distance and Area under the Curve

Please see attached file for full problem description. 7. Figure 5.4 shows the velocity, v, of an object (in meters/sec). Estimate the total distance the object traveled between t = 0 and t = 6. We can estimate this using 1 second intervals. Since the velocity is increasing on the interval from t = 0 to t = 6, the lo

Double Integration : Joint PDF, Marginal and Conditional Densities

Please see the attached file for the fully formatted problems. Suppose that X and Y are continuous random variables with the joint probability density function k(x+y) for 0<x<1,0<y<2 f(x,y) = 0 otherwise (a) Find k, E(X), E(Y), V(X), V(Y), and Cov(X, Y). (b) Are X and Y independent ? (c)

Limits, L'Hopital's Rule and Integrals

1. Evaluate 2) 2. Differentiate the function f(x) = ln(2x+3) 3. Find lim x&#61664;&#8734; (e2x / (x + 5)3). Apply L'Hopital's rule as many time as necessary, verify your results after each application. 4. Evaluate &#8747;xsinh(x)dx See attached file for full problem description.

Integration

1.R is the region that lies between the curve y = (1 /( x2 + 4x + 5) ) and the x-axis from x = -3 to x = -1. Find: (a) the area of R, (b) the volume of the solid generated by revolving R around the y-axis. (c) the volume of the solid generated by revolving R round the x-axis. 2.Evaluate: &#8747; sinh6 x cosh xdx.

Integration : Area of a Bounded Region

Please help with the following problem. Provide step by step calculations for each. The average value of f(x) = 1/x on the interval [4, 16] is (ln 2)/3 (ln 2)/6 (ln 2)/12 3/2 0 1 none of these Find the area, in square units, of the region b

Stokes Theorem

Stokes Theorem. See attached file for full problem description. Use Stokes Theorem to evaluate....

Integration: Finding Work Done by Gas and Moment of Mass

4. A quantity of gas with an initial volume of 1 cubic foot and a pressure of 2500 pounds per square foot expands to a volume of 3 cubic feet. Find the work done by the gas for the given volume. Assume that the pressure is inversely proportional to the volume. 6. Find moment of mass M_x, M_y, and center of the mass for the la

Finance/Accounting Problems

10. Stock Values. Integrated Potato Chips paid a $1 per share dividend yesterday. You expect the dividend to grow steadily at a rate of 4 percent per year. 1. What is the expected dividend in each of the next 3 years? 2. If the discount rate for the stock is 12 percent, at what price will the stock sell? 3. What is the exp