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# Integrals

### Integration: The Limit of a Sum

Find by the method of summation the value of : a) The integral (from 0 to 1) of the square root of x. (dx) b) The integral (from 1 to 4) of 1 divided by the square root of x. (dx) Please view the attachment for proper formatting.

### method of summation the value of the integral is sin(x) and cos

Integration as the limit of a sum (II) Find by the method of summation the value of: a) The integral (from 0 to 1/2*pi) of sin(x). b) The integral (from a to

### Find by the method of summation the value

Integration as the limit of a sum (I) Find by the method of summation the value of:-

### Cauchy's Formula Trigonometric Function

See Attachment for equation We know that sin z and cos z are analytic functions of z in the whole z-plane, what can we conclude about *(see attachment for equations)* in the first quadrant

### Integration as the limit of a sum

Find the integral by the method of summation the values of :- (a) integral of e^(-x) where the range of integration is from a to b. (b) integral of e^(kx) where the range of integration is from a to b.

### Mathematica integration

I am new to Mathematica and did derive the answer to the following but I can not get the information as to the steps taken to derive it. I am using the trapezoidal and Simpson's rules to evaluate S20 x2 dx ( the S should be the variant symbol) Compare with exact value. My answer is 8/3 but I can not get Mathema

### Z-Transforms & Grad, Div, Curl

Please see the attached file for the fully formatted problems.

### Volume of a solid..

Please see word attachment for clearer view of the problem. Volume: Find the volume of the solid generated by revolving the region bounded by the graphs of y = xe^-x, y = 0, and x = 0 about the x-axis.

### Calculus Functions to Evaluate an Integral

Please help with various Calculus questions. You do not need to show your work for this one because I would simply like to compare your answers with mine so that I am sure that I did everything correct on mine. Please just write your exact answer after each number. I will know which problems I will have to study in detail w

### Improper Integrals surface area problem..

Use l'hopital's rule if needed. Show all work step by step. Use proper notation please. Surface Area: The region bounded by [(x-2)^2]+y^2=1 is revolved about the y-axis to form a torus find the surface area of the torus. PLEASE show or explain step by step process.

### Positive Integral Exponents

(-x(t^2))*((-2(x^2)t)^4)

### Positive integral exponents variables

[2(a^5) * 3(a^7)(b^3)]/[15(a^6)(b^8)]

### Work in Joules of a Vertical Cross Section

A trough is 2 meters long, 2 meters wide, and 2 meters deep. The vertical cross-section of the trough... (See attached)

### Finding the centroid of a 2D shape.

Find the centroid of a two dimensional shape that is formed by the intersection of the lines: y = x-3 and y = x^2

### Partial Derivative and Double Integral

The problems are attached 1 -5 based on Chapter Partial Derivative - (Maximum & Minimum Values and Lagrange Multipliers 1. Locate all relative maxima, relative minima, and saddle points of the surface defined by the following function. 2. Consider the minimization of subject to the constraint of (a) Draw the

### Revolution of integrals - asteroids

If f(θ) is given by: f(θ)=6cos^3θ and g(θ) is given by: g(θ)=6sin^3θ Find the total length of the astroid described by f(θ) and g(θ). (The astroid is the curve swept out by (f(θ), g(θ)) as θ ranges from 0 to 2pi)

### Revolutions of integrals - torus

The circle x=acost, y=asint, 0≦t≦2pi is revolved about the line x=b, 0<a<b, thus generating a torus (doughnut). Find its surface area. Area if the torus:_____________.

### Rotation of Solid Integrals

Find the volume of a solid generated by revolving about the x-axis the region bounded by the upper half of the ellipse *See attached for equation* and the x-axis and thus find the volume of a prolate spheroid. Here a and b are positive constants, with a<b Volume of the solid of revolution: Please see attachment for det

### Integral from 0 to Infinity

Show that the integral from 0 to infinity of (t^n e^-t dt) = n!

### Volume of a Solid Integration

See attached for Diagram The base of a certain solid is the area bounded above by the graph of y=f(x)=16 and below by the graph of y=(gx=36*. Cross sections perpendicular to the x-axis are squares. See picture above. Use formula (see attachment) to find the volume of the solid.

### Evaluating integrals

Evaluate the integral from 0 to infinity of (e^(-at) - e^(-bt))/t dt for a,b > 0

### Integration - Volume of Rotation

Find the volume of the solid obtained by rotating the region bounded by the given curves: y=1/x^6, y=0, x=4, x=8 about the "y" axis

### Evaluating integrals BrainMass Expert explains

Evaluate the integral from 0 to infinity of (sin (xy))^2 / x^2 dx

### Volume of a Solid of Rotation

Find the volume of the solid formed by rotating the region inside the first quadrant enclosed by: y= x^4 y= 125x about the x-axis. I am more concerned with understanding than the answer. Thanks for your help.

### Integrate the differential equation

(x^2 + 3y^2) dx - 2xydy = 0 Integrate the differential equation. Complete step by step work must be shown and reduced into lowest terms.

### Integration Functions Solved

If f(x) = int_{1}^{x^{2}} t^2dt then f'(x)= then f'(5)=

### Evaluate the integral

The following expression describes the total electric current to pass in the circuit please see attached

### Find the area between the curves

Find the area between the curves y = x2 + x + 1 and y = 2x + 7 From x = -2 to x = 3 Sketch the curve and indicate the interval and the enclosed area

### Integrating to Find an Area of a Region

Decide whether to integrate with respect to "X" and "Y", then find the area of the region. x+y^2=42, x+y=0.

### Integral domain

There is integral domain with exactly six elements. Disprove or Prove