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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.

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

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

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

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

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

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....

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

Contours and the Cauchy Integral Formula

Let C be the boundary of the square of side length 4, centered at the origin, with sides parallel to the coordinate axes, and traversed counterclockwise. Evaluate each of the attached integrals.

Approximating an integral

Approximate the integral by: a) first applying Simpson's Rule b) then applying the trapezoidal rule See attached file for full problem description.

Convergence and infinite series

1.) Find the interval of convergence of the series &#931; (for n=0 to &#8734;) (4x-3)^(3n)/8^n and, within this interval, the sum of the series as a function of x. 2.) Determine all values for which the series &#931; (for n=1 to &#8734;) (2^n(sin^n(x))/n^2 converges. 3.) Find the interval of convergence of the series &#931

Show that the two iterated Riemann integrals of the given function of two real variables are unequal to each other, and that the absolute value of the function is not Lebesgue integrable.

Let f be the following function with domain C = [0, 1] X [0, 1] (in two-dimensional Cartesian space): f(x, y) = 0 on the line segments x = 0, y = 0, and x = y f(x, y) = -1/(x^2) if 0 < y < x <= 1 f(x, y) = 1/(y^2) if 0 < x < y <= 1 Compute each iterated Riemann integral of f on C (by integrating first over x and then

Sequences and Improper Integrals

1.) Show that the functions f1(x)=5^x, f2(x)=5^(x-3), ans f3(x)=5^x + 3^x all grow at the same rate as x approaches infinity. 2.) Determine whether each integral converges or diverges. a.) integral from 0 to 2 of (dx)/(4 - x^2) b.) integral from 0 to infinity of (5 + cosx) e^(-x)dx c.) integral from 0 to in