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# Integrals

### Evaluate derivatives of several functions

Evaluate: ∫ sinh6 xcosh xd x Given ?(x)= csch^-1 1 /x2 find ?'(x) Given ?(x)=log10x find ?'(x).

### 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

### Let I be the set of all integers and let m be a fixed positive integer. Two integers a and b are said to be congruent modulo m-symbolized by a is congruent to b (mod m) - if a - b is exactly divisible by m, i.e., if a - b is an integral multiple of m. Show that this is an equivalence relation , describe the equivalence set, and state the number of distinct equivalence sets.

Topology Sets and Functions (XLVII) Functions Let I be the set of all integers and let m be a fixed positive integer. Two integers a and b are said to be congruent modulo m-symbolized by

### Integration by parts explained in this solution

What is the integral of x ln x dx? ∫xlnx dx

### Areas between a curve and the x-axis

What is the area between f(x) = x^3 - 4x and the x-axis for the interval x = -2 to x = 2?

### 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

### Conic Section in Polar Coordinates

#7,8 and 9 Please see attached file.

### 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

### Surface Integrals

Find the surface integral (double integral over S) E dot dS, where S is the cylinder, x^2 + y^2 = 4, z is greater than or equal to 2 and less than or equal to 5, and the vector field F is F(x, y, z) = (0, 0, z^2)

### Vector Fields and Surface Integrals

Find the flux of the vector field F(x, y, z) = (y, 0, z2) out of the unit sphere S. In other words, find the surface integral ∫∫S (y, 0, z2) * dS, where the sphere S is oriented by the outward normal. Let S be the cylinder x2 + y2 = 1, 0 ≤ z ≤ 6. Find ∫∫S (x4 + 2x2y2 + y4)2 dS.

### Calculus: Evaluating an Integral

Let C be the curve represented by the equations x = 2 t , y = 3 t^(2). Evaluate the integral (0 <= t <= 1) / l (x - y)ds . l / C

### Cantor Sets

1. Show that the Cantor function c: [0, 1] → [0, 1] is continuous. To do this, I know I need to use the fact that c is monotone, but I'm having difficulty from there. 2. Compute ∫c, where c is considered to be an element of L+(R). (let c(x) =0 for x not in [0, 1]) Here, c is the Cantor Function and L+(R) consists

### 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.

### Linear Operators, Inner Products and Adjoints

We are studying an inner product spaces. See attached file for full problem description. Let V be a C-space of all complex valued polynomials with an inner product.... (i) Let p be a polynomial and let Mp: V-> V be a linear operator that is given by Mp (q) :=p&#8901;q. Show that operator Mp has an adjoint and find it. (i

### 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