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Integrals

Taylor polynomial, Lower and Upper sums

2a. Find the second order Taylor polynomial for f(x) = x^(1/3) about x = x0, x0 > 0. 2b. Show that the function G(x) is differentiable at x0 and find G'(x0). 3a. Find an expression for the lower sum L(f,D) and upper sum U(f,D). 3b. Determine the lower integral and upper integral. Please refer to the attached images f

line integral solving process

Please show the steps in solving process. Parts of the questions are given below to give a quick idea about them; please refer to the attachment for complete and all the questions. (1) Evaluate the line integral. (2) Let C be a circle of radius 1 centered at the point (x,y) = (1,0). Consider the vector field F(x,y) = xi +

Integration and Differentiation

1. Evaluate the integrals. a. Integral of cot cubed root of x / cubed root of x^2 dx b. Integral of cos^2x / sinx dx c. Integral of cosx sinx / cos^2x-1 2. Solve the differential equation subject to the given conditions. y'' = 6 e^2x ; y = -3 and y' = 2 if x = 0 3. Find f'(x) for each of the follo

Cauchy-Goursat Theorem

Given: Integral from zero to infinity of cos(x^2) dx = integral from zero to infinity of sin(x^2) dx = 1/2 * (square root of pi/2). These can be evaluated by considering cos(x^2) = Re(e^ix^2) and sin(x^2) = Im(e^ix^2). 1.) Integrate the function f(z) = e^i(z)^2 around the positively oriented boundary of the sector 0 <= r

Improper Integrals

1.) Use residues to evaluate the improper integral from 0 to infinity: 1 / (x^2 + 1)^2 dx 2.) Use Jordan's Inequality to evaluate the improper integral from -infinity to infinity: (x^3 sin ax) / (x^4 + 4) Thank you for your assistance.

Calculating Integrals

Evaluate the integral. *** Integral is attached *** Note: Integral is followed by dx This *should be* "integral of 4x^2 times the fourth root of (8 + 4x^3) times dx" and each of the smaller numbers in the choices below are the exponents. Second Attachment should be here! a. -8/3(8+4x^3)^-3/4 + C b. 4(8+4x^3)^5/4

Fundamental Theorem of Line Integrals

a). Show that the line integral ∫_C▒〖ysinxdx-cosxdy〗 is independent of the path. b). evaluate the integral in part (a) along the line segment from (0, 1) to (π,-1) c). Evaluate the integral ∫_((0,1))^((π,-1))▒〖ysinxdx-cosxdy〗 using Theorem 16.3.1 and confirm the value is the same as that obtained in part

Stable Steady-State Value

Solve the following linear, first-order differential equations and ensure that the initial conditions are satisfied. Show whether or not the steady-state solutions are stable. (a) 10y' = 5y and y(0) = 1. The answer is y(t) = e^(1/2t), but having trouble arriving at that answer (b) 4y' - 4y = -8 and y(0) = 10. The answer

Integration of trigonometric functions

I am supposed to find or evaluate the integral of cos^(2)XsinXdx I was thinking that i should use integration by parts but I am not sure what values to use for U, and dv...

contradiction to the condition

Let (X, E, u) be a measure space with u a non-negative measure. Suppose that 1. f : X -> R is measurable 2. f (x) >= 0 a.e. with respect to u. 3. integral (over X) f du = 0 Prove that f (x) = 0 a.e. with respect to u. note: E = Capital Sigma u = lowercase mu

Trapezoidal Rule Word Problems

Please show ALL work! A bacteria population grows at a rate proportional to its size. Initially the population is 10,000 and after 5 days it's 30,000. What is the population after 10 days? How long will it take for the population to double? A solid S is generated by revolving the finite region bounded by the y-axis

derivative and the slope of a curve

In this explanation, I am being asked to discuss the relationship between the slope of a secant line, the slope of a tangent line and the derivative AND in addition, I must explain the relationship between the area of a finite number of rectangles under a curve and an infinite number of rectangles under a curve and the definite

Differential Equations and Indefinite Integrals

1. Find the solution of the differential equation dy/dx=x^3 -x, with the initial condition y(0) = -3. 2. Find the solution of the differential equation dy/dt=t^2 / (3y^2), with the initial condition y(0) = 8. 3. Find the solution of the differential equation dy/dx=(x=2)y^(1/2), with the initial condition y(0) = 1. 4. Fi

Calculaion of work done on a spring

1) Find the arc length of the curve f(x)=x3/2 - 1 over the interval [0,1] 2) Find the arc length of the curve f(x)=ln(cos x) over the interval [0,pi/4] 3) Find the arc length of the curve f(x)=1/6 x3 + 1/2 x - 1 over the interval [1,2] The Next TWO Problems Refer To The Paragraph Below. This is a two step module. First you

Defining a Function: Example Problem

A rational number r = p/q, where p, q are in Z, is said to be properly reduced if p and q (q > 0) have no common integral factor other than +1 or -1. Define the function f as follows; f(x) = q , if x = p/q, properly reduced. f(x) = 0 , if x is irrational. Prove that for every real number x, f fails to be bounded at x.

Divergence of Improper Integrals

The integral from 0 to 1 of 6/x-1 dx If I am correct, the answer guide says this one converges but my answer is going to infinity which is diverges. I need to see your steps to compare with mine please.

Convergence - Series Expanision for Integral

Obtain a series expansion for the integral ∫_0^(1/2)=1/((1+x^4)) dx and justify your calculation. Please see attachment for full problem. This is for an analysis class, so i need to understand how you get the series expansion, so please give details.

Finding the Integral: Substitution

Find the integral using each method. Integral x * sq. rt 4 + x dx a) trigonometric substitution b) Substitution: u^2 = 4 + x c) Substitution: u = 4 + x d) Integration by parts: dv = sq. rt. 4 + x dx Thanks.

Evaluate the Double Integral Over the Region: Example Problem

Evaluate the double integral over the region R. Double int. over R (x)(sq. rt. 1-(x^2) dA: R=[(x,y):0<_x<_1, 2<_y<_3 If I set up the integral f(x,y) dy dx isn't the integral with respect to y equal to zero? I am left with the integral from 0 to 1 of f(x,y) dx then I am having issues actually integrating.

Integration of Polynomials and Trigonometric Functions: 25 Problems

Integrate the following functions: 1. (4x^2-8x+1)dx 2. (9t^2-4t+3)dx 3. (2t^3-t^2+3t-7)dt 4. (1z^3-3z^2)dz 5. (4z^7-7z^4+z)dz 6. (3 square root u + 1square root u)du 7. (square root u^3-12u^-2+5)du 8. (2v^54+6v^14+3v^-4)dv 9. (3v^5-v^53)dv 10. (3x-1)^2dx 11. (x-1x)dx 12. x(2x+3)dx 13. (2x-5

Solving second order differential equation

Find the general solution of the second order differential equation y'' - y' - 6y = e^-3x This one is quite long winded, and I am pretty sure that I am getting yh right but can't seem to get close to yp. I think this is a D.R.A.E?

Concepts of Integration

Please solve the following integration application problems. Please provide steps to arrive at final answer. 1. Find the solid of revolution for f(x) = x^2 and g(x) = 1/2 x^3 for x∈[2, 3] 2. What are the coordinates of the centroid for f(x) = x &#8704;∈[0, 2] 3. Integrate x^2 + 3x - 4 / x 4. How much total work is ex

Definition of a Derivative

Calc Proofs 1) Using the following two functions X and X^2 develop their derivatives using the Definition of a Derivative for three values of "h" h = .1 h = .01 h= .001 and then repeat the calculation in the limit as h->0 2) Using the two functions above, show that the Finite Sum approximation of the area

Mathematics - Average Value of the Function

Find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value. Round your answers to four decimal places. f(x)=16-x^2, [-4,4] (x,y)=(_______)(smaller x-value) (x,y)=(_______)(larger x-value) Thanks

Mean Value Theorem for Integrals

Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function over the given interval. (Round you answer to four decimal places. Enter your answers as a comma-separated list.) f(x)= 8 times sq.rt x, [4,9] c=______ thanks

Reimann integrable problem

If f is a reimann integrable function on [a,b], and if [c,d] is a subset of [a,b], prove that f is reimann integrable on [c,d] hint: if P is any partition of [c,d], P can be extended to a partition P* of [a,b] with ||P*|| <= ||P||. Show that U(f,P) - L(f,P) <= U(f,P*) - L(f,P*)

Two Reimann Integrable functions

Are these functions Reimann Integrable? I am just learning this topic, so my description may not be accurate. A function is Reimann Integrable if it's Upper Darboux Sums and Lower Darboux suns are equal. Or stated another way, if U(f, P) - L(f, P) < e The two functions are piecewise functions. 1) f(x) = { 0 when x =

Differential Equation Problem

Part I Check that N(t) = t/(1 +ct) is a solution of the differential equation dN/dt = N^2/t^2. Treat c as an unspecified constant. Part II Use that N(1) = -1 to find c. Then give the solution N(t) corresponding to this initial condition.