Share
Explore BrainMass

Integrals

An Integral is a function, F, which can be used to calculate the area bound by the graph of the derivative function, the x-axis, the vertical lines x=a and x=b. It is commonly written in the following form:

Int_a->b_f(x)

where,

Int is the operation for integrate

a and b represent the vertical lines bounding the area

f(x) is the derivative function

Thus for the simple function y=3x^2, we can integrate in the following way:

Y = 3x^2

F = int(3x^2)

F = (3x^3)/3

F = x^3

If the x values were 1 and 2, then the area under the curve would be:

F(2)-F(1) = 8-1 units.

However, finding the derivatives is usually never as easy as the above example. Instead there are many different methods to find the integral of complex functions. For example, one can integrate by parts to find the integral of a product of functions. Consider the following function:

f(x) = x*e^2x

We can split the function into two parts:

u = x

dv = e^2x

Integrating by parts has the following formula:

int_u*dv=u*v-int_v*du

Thus we can calculate:

du = 1

v = (1/2)*e^2x

Therefore, the integral is:

Int_x*e^2x = x*(1/2)*e^2x – int_(1/2)*e^2x*(1)

Int_x*e^2x = x*(1/2)*e^2x – int_(1/4)*e^2x + c

From this example it can be seen that finding the integral is not always straightforward. Thus, understanding the complexities and the multitude of different rules to integrate a function is crucial for the study of calculus.

maths calculus help

sketch the curve for y = sin(X) between x = 0 and x = 2pi, then find the total area enclosed by the curve y = sin x and the X axis between x = 0 and x = 1.7pi I have got the sketch a graph bit sorted i just can't work out the rest. if you could provide full workings and method that would be great also ∫ (5x^2 + sqrt(x) - 4/

Real Analysis - definite integral

I have been doing Integrals, and have been successful up until this one. Please note at the end of the equation the "ds", rather than "dx" which I believe results in a different method of solving. The integral I need to find is in the picture attached.

Integrals with limits

Solve the integral -x dx with limits from 0 to x I am actually solving the differential equation, y'=-xy y(0)=5, but I need to see the above integral solved from 0 to x to reassure myself that my other problem is working correctly.

Evaluation of Indefinite and Definite Integrals

(a) Find the indefinite integrals of the following functions (i) f(t)=27 cos(9t)+4e^(-12t) (ii) g(x)=(14+45x^3)/x (x >0) (iii) h(u)=(sin)^2 (1/4u) (b) Evaluate integration[3, 7]x(5x^2+9)dx (c) (i) Write down a definite integral that will give the value of the area under the curve y=x^3cos(2/5 x) between x=1/2pi and

Separation of variables, sine and cosine expansion.

Full equations are shown in the attached file. 1. Let f(x) = x(1-x), 0 < x < 1 Find: a. cosine series expansion of f(x). b. sine series expansion of f(x). Sketch the extended function for series in (a) and (b). 2. Solve the wave equation: [attached] with boundary conditions: u(0,t) = u(1,t) = 0 and initial condition:

Solving: A Single Integral

Please solve: K = Integral from zero to infinity of x^(-a)dx/[1+2 x cos(theta) + x^2] Please see the attached PDF document for the integral that needs to be solved.

Showing that convergence is not uniform

Let fk (x) = kxe - kx, k = 1, 2, 3, ... It can be shown that the sequence {fk} infinity, k = 1, converges to 0 Pointwise on [0, +oo) but convergence is not uniform on [0, +oo). a. Show that convergence is not uniform on [0, 1] either. Thus the sequence does not converge in the normed vector space (C ([0, 1]) , II·II in

contour integration

Do the integral (0 --> infinity) using contour integration: J(a,b) = dx sin(bx)/sinh(ax)

Vector Calculas and the Applications

1. (i) Find the total derivative and the Jacobian for the following change of variables: x = acos(uv) y = bsin(vw) z = xexp(-uw) (ii) Simplify the equation: see attached using the change of variables: see attached See attached. 2. Find the Jacobian Jpar of the coordinates transformation for the parabolic coordin

Solving Integrals

1. Write any integral from the next page that requires a constant of integration after calculating the antiderivative. (Include ALL notation. Do not evaluate the integral here.) 2. Write any integral from the next page that requires the Fundamental Theorem of Calculus to evaluate an area under a curve. (Include A

Integrating the Poisson Kernel

I have been trying several methods to try to integrate the following function: Integral[Pa(theta)*d(theta)] *Recall that Pa(theta) = (1-a^(2))/(1-2a*cos(theta) + a^(2)) and see the attached file for the proper notation. I substituted 1/2(z+1/z) for cos (theta) and reduced the numerator to (az-1)(a-z) in hopes that there wo

Contour and Fresnel Integrals

We want to calculate the integral of the function x*sin(x^4), from 0 to infinity. We may think that this is similar to a Fresnel Integral (sin(x^2)). In that case, we would set z=e^(iz)^2, and then integrate over the special contour with regions I: 0 to R, II: theta = 0 to pi/4 and then finally to region III: R to 0 along e

Rayleigh Ritz method

Use the Rayleigh-Ritz method to find two successive approximate solutions to the extremum problem associated with the functional: J[y]=Integral from 0 to 1 of (y'')^2-y^2 dx, y(0)=0, y'(0)=0, y(1)=1, y'(1)=1, using the trial function Y_0(x)=a_3 x^3+a_2 x^2+a_1 x+a_0, where a_0, a_1, a_2, and a_3 are chosen so that Y_0(x)

Rayleigh Ritz Method

Use the Rayleigh-Ritz method to find three successive approximate solutions to the extremum problem associated with the functional: J[y]=Integral from 0 to 1 of (y')^2+xy^2+2x^2y dx, y(0)=0, y(1)=1, using the trial functions Y_0(x)=x, Y_1(x)=x+c_1 x(1-x) and Y_2(x)=x+c_1 x(1-x)+c_2 x^2(1-x)^2.

Maximal Interval of Existence of the Initial Value Problem

Please solve the following initial value problem: dx/dt = x^2 - 4, x(0) = 0. and find the maximal interval of existence of the solution. We solve the initial value problem using separation of variables. We use partial fractions to solve for the integral of 1/(x^2-4).

Numerical integration using Simpson and Gaussian Quadrature

With the specified accuracy integrate the function using Simpson's one-third rule and Gaussian quadrature: Int_0^3 Int_{-1}^2 Int_1^4 e^x y^3 z^2 dxdydz, where Int_a^b f(x)dx denotes the integral of the function f from a to b. See the attached file.

Maximal Interval of a Dynamical System

Please help solve the following nonlinear system: For every xo in R, solve the initial value problem; dx/dt = sin x , x(0) = xo i) find the maximal interval of existence ii) find the dynamical system defined by the solutions.

Calculating Velocity and Distance

I'm having trouble with this word problem. Please show all work. 8. Ignoring resistance, a sailboat starting from rest accelerates at a rate proportional to the difference between the velocities of the wind and the boat. (a) The wind is blowing at 20 knots, and after 1 half-hour the boat is moving at 10 knots. Wr

Numerical Analysis - Simpson's Rule

a) Explain how we arrive at the formula for Simpson's rule (standard formula) using the Lagrange Interpolating Polynomial of degree 2. Ignore the error term, and do not compute any integral. b) We define the Composite Simpson's Rule by splitting the interval [a,b] into smaller sub-intervals, applying Simpson's Rule on those s

Finding Solutions of Congruence

Question: Find the solutions of the congruence 12x^2 + 25x = 10 (mod 11). [Hint: Show the congruence is equivalent to the congruence 12x^2 + 25x + 12 = 0 (mod 11). Factor the left-hand side of the congruence; show that a solution of the quadratic congruence is a solution of one of two different linear congruence's.]

Euler-Lagrange Expression

Write down the Euler-Lagrange expression for the function. Let f (x,x',x"). Be a function of independent variables x, its first derivative x', and its second derivative xo. Write down an Euler-Lagrange expression for this function.

Finding the Minimum of a Functional

For which curve u(x) does the functional L[u], defined as the integral of F[x, u, u'] = (1/2) (u')^2 + u u' + u + u', attain a minimum when the values of u(x) are not specified at the end points of the interval of definition [a, b]?

Calculating Velocity, Integrals, and the Sum

Please see the attachment for 3 questions I have. 3 homework questions. Please help. 8. The velocity of the flow of blood at a distance from the central axis of an artery of radius is where is the constant of proportionality. Find the average rate of flow of blood along a radius of the artery. (Use 0 and

Find the indefinite integral and check

1. Find the indefinite integral and check the result by differentiation: a) integral (x^3 - 10x - 3) dx b) integral (t^2 - cost) dt 2. Evaluate the definite integral by the limit definition: integral on the interval -2 to 1 (2x^2 + 3) dx