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Finding a centroid

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

Revolutions of integrals - astroids

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 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


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


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.


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


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.


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


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

MATLAB Programming

Develop a program (M-File) called 'integrate' that will perform a first-order numerical approximation, yi(t), of the running integral with respect to time of an array of experimental data y(t). The M-File must also perform another first order approximation, yi2(t), of the first integral resulting in a double integration of the

Differential Equations

Please see the attached file for the fully formatted problems. y(t) + S t-->0 (t-tau)y(tau) dtau = e^t


Determine the indefinite integral: z = (Integral Sign or "Long S")20xe^-4x dx. I got z = 5xe^-4x + 5/4 e^-4x + C - Does that seem right?

Simpson's Rule

Find the integral of e^-x dx from x = 0 to 1 with Simpson's rule using 10 strips.

Evaluate the Surface Integral

Please see the attached file for the fully formatted problems. Evaluate the surface integral SSs x dS, where S is part of the plane x = 2y = 3z = 6 that lies in the first octant.

Green's Theorem : Evaluating a Line Integral

Please see the attached file for the fully formatted problems. Use Green's Theorem to evaluate the line integral Sc xy dx +x^2y^3 dy where C is the triangle with vertices (0,0), (2,0) and (2,2).

Gradients : Find a Function and Evaluate an Integral

Please see the attached file for the fully formatted problems. (a) Find a function f so that grad(f) = yi + (x + 3y^2)j (b) Use part (a) to evaluate Sc grad(f) dt where C is the path starting at (0,2) goes down the y-axis to (0,0), along the x-axis to (2,0).

Iterated Integral

Please see the attached file for the fully formatted problems. Evaluate the iterated integral S 1-->0 S z--> 0 S x+z --> 0 x dydxdz

Polar Coordinates

Please see the attached file for the fully formatted problem. By changing to polar coordinates, evaluate SSR (x+y) dA where R is the disk of radius 4 centered at the origin.


How do you integrate: [a^x exp(-a) ] / x ! (The ^ represents 'to the power of ' so a^x implies a raised to the power x)


I've included the problem as a JPEG . Thank you