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Integrals

Area and volume using multiple integrals

1 Find the area of the part of the surface 2z = x^2 that lies directly above the triangle in the xy-plane with the vertices at (0,0),(1,0) and (1,1). 2 Find the volume of the region in the first octant that is bounded by the hyperbolic cylinders xy = 1, xy = 9, xz = 4, yz = 1, and yz = 16. Use the transformation u = xy, v =

Indefinite Integrals : Partial Fractions

Use the method of partial fractions to evaluate the indefinite integral (Let u= ln[x]) (integral) (7+11*ln[x]^2)/(x*ln[x]^3+x*ln[x]) keywords: integration, integrates, integrals, integrating, double, triple, multiple keywords : find, finding, calculating, calculate, determine, determining, verify, verifying, evaluate, e

Volume of a Solid of Revolution

Identify the definite integral that represents the volume of the solid formed by revolving the region bounded by y=x^3, y=1, and x=2 about the line y=10.

Indentify definite integral

Identify the definite integral that represents the area of the region bounded by the graphs of y=x and y=5x-x^3

Area of surface formed by revolution

Identify the definite integral that represents the area of the surface formed by revolving the graph of f (x)= x^3 on the interval [0,1] about the y-axis.

Description of Simpson's Rule

A function f is given by the following table: x= 0 1 2 3 4 f(x)= 9 2 2 8 5 Approximate the area between the x-axis and y=f (x) from x=0 to x=4 using Simpson's Rule.

Indefinite integral

Use the method of partial fractions to evaluate the indefinite integral. (hint: let u=ln[x]) (8+9*ln[x]^2)/(x*ln[x]^3+x*ln[x])

Work Done

A force of 20 pounds stretches a spring 3/4 foot on an exercise machine. Find the work done in stretching the spring 1 foot. keywords: finding, find, calculate, calculating, determine, determining, verify, verifying, evaluate, evaluating keywords: integrals, integration, integrate, integrated, integrating, double, triple,

Convergence or Divergence of an Integral

Evaluate the integral or determine that it diverges: (integral from positive infinity to 0) x*e^(-x/2)*dx keywords: integrals, integration, integrate, integrated, integrating, double, triple, multiple, improper

Holomorphic Function and Taylor Expansion

Let f(z) be holomorphic in |z|less than R with Taylor expansion f(z)=sum(a_nz^n) and set I_2(r)=1/2pi(integral from 0 to 2pi of|f(re^itheta)|^2 d(theta), where 0<=r<R. Show that a) I_2(r)=sum(n=0 to 00)|a_n|^2r^2n b) I_2(r) is increasing. c) |f(0)|^2<=I_2(r)<=M(r)^2, with M(r)=sup_|z||f(z)|

Convergence or Divergence of an Improper Integral

6. Determine the divergence or convergence of the given improper integral. Evaluate the integral if it converges. integral to the power of 4 sub 3 1 / (square root x - 3) dx 7. Use integration by parts to evaluate the given integral. integral x sec^2 x dx

Derivatives and Integrals : Area and Volume of Solid

Explain why the derivative function of the function g(x) = x is equal to 1 on the interval (0,&#8734;), equal to 1 on the interval (-&#8734;,0), and undefined at 0. [Hint. Sketch the graph of g.] Consider the region R bounded by the curve xy =3 and the lines x I and x =4. Set up the integrals (do not evaluate) that give the

Stokes' Theorem : Curls and Surface Integrals

Let F = (2x, 2y, 2x + 2z). Use Stokes' theorem to evaluate the integral of F around the curve consisting of the straight lines joining the points (1,0,1), (0,1,0) and (0,0,1). In particular, compute the unit normal vector and the curl of F as well as the value of the integral:

Integration Word Problems : Rate of change

Suppose that a tank initially contains 2000 gal of water and the rate of change of its volume after the tank drains for t min is '(t)=(0.5)t)-30 (in gallons per minute). How much water does the tank contain after it has been draining for 25 minutes? keywords: integration, integrates, integrals, integrating, double, triple, m

Volume of a solid

The region R is bounded by the graphs of x-2y = 3 and x=y^2. Set up (but not evaluate) the integral that gives the volume of the solid obtained by rotating R around the line x=-1.

Integrals and Average Sums

Please see the attached file for the fully formatted problems. keywords: integration, integrates, integrals, integrating, double, triple, multiple

Solids

The region bound by the circle (x-a)^2 + y^2 = a^2 is revolved around about the y-axis to generate a solid 1 find the volume 2 find the surface area

Integration and Sums of Rectangles

1. Given f &#8242;(x) = ex + (4/3)x^(-2/3) , find f (x) if f(1) = e. 2. Given f (x) = x2 + x +1 (a) Approximate the area between the curve of f and the x-axis on the interval [0,2] using 4 rectangles and right point sums. (b) Find the EXACT area between the curve of f and the x-axis on the interval [0,2] by using area a

Definite Integrals

1) 3x+6/(x^2=4x=5)^2 dx --This is a definite integration problem evaluated at b (1) and a (-1) 2) ln x/x dx --Evaluated at b (e) and a (1) 3) x^1/2 + 3x^1/3 + 3 dx --This is an indefinite integration problem. 4) e^3x + 3/x dx --Indefinite 5) e^-5x + e^5x dx --Definite-Evaluated at b (1) a (-1)

Plane Triangular Surface and Stokes' Theorem

4. Consider the plane triangular surface formed by the intersection of the plane x/A + y/B + z/C = 1 (A, B, and C all positive), with outward pointing normal, ie the normal pointing away from the origin. Verify Stokes' Theorem for the vector field F = (x + y) + (2x &#8722; z) + (y + z) by performing the surface integral a

Surface Integral of a Paraboloid of Revolution

Let S be the closed surface of the paraboloid of revolution z = ±(4 &#8722; x2 &#8722; y2 ) where &#8722;2 x, y +2. Evaluate the following surface integral directly and then by using the divergence theorem; where R is the position vector to a point on the surface and is the outward pointing normal at that point. See att