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Triple Integrals : Finding Volume of Solids with Boundaries

1) Evaluate the triple integral e^(1-(x^2)-(y^2)) dxdydz with T the solid enclosed by z=0 and z= 4-(x^2)-(y^2) 2) Find the volume of the solid bounded above and below by the cone (z^2) = (x^2) + (y^2), and the side by y=0 and y= square root(4-(x^2)-(z^2))

Quick Calculation of Laplace Integral

Please see the attached file for the fully formatted problem. Construct the quickest method to calculate the Laplace Integral. I = S e^(-x^2) dx infinity --> infinity


Integral x-1, divided by x to the 3rd + x squared to dx. x-1 ----- X to the 3rd + X to the 2nd ( all dx)


For problem #1, its the integral from o to infinity (the symbol for infinity for that problem was cut off)

Evaluating integrals

I'm taking a DE calculus class and I'm having problems figuring out the logic in solving some of the problems. The given integral is improper because both the interval of integration is unbounded and the integrand is unbounded near zero. Investigate its convergence by expressing it a sum of two intergrands-one from 0 to 1 an

Double Integration

Evaluate the double integral Transform the double integral of (i) using plane polar coordinates Show that the 3 x 3 determinant See attached file:

Evaluate integrals

Please see the attached file for the fully formatted problems. Evaluate the following integrals. S (4x^3 -2x - (2/x^3) dx S (1/2x^1/2) dx 1-->0 S ln x dx

Integration: Integration by Parts

I am trying to integrate e to a variable power times sin or cos using integration by parts, but I seem to be going in circles. How is this problem solved? The trick for solving e times sin or cos is shown using the example Se^x*sinxdx.

Integration of cos^2(x)

How do I integrate cos^2(x)? Please help me with this and include explanations so I can understand it.

U substitution

U substitution is explained using the example S4x(x^2+1)^5dx without and with limits of integration.


(a) Find the integral between o and infinity (upper)of e^-x^2 dx . Use the above to prove that T(/2)= sqare root of pi where T represents the gamma function (b) Find the integral of x^3.e^-x^2 dx between the boundaries 0 and infinity (upper) thank you

Integration of a function

(a) let f:[0,1] ---R be the function f(x) = { x when x is an element of rational numbers {-x when x is not an element of rational numbers Prove that f is not integrable on [0,1] but |f| is integrable (b) Find the limit as x goes to 0 of 1/x the integral of e^t^2 dt between the boundaries 0 and x, x bei


(a)Let f:[a,b] ---R be continuous and f(x)>= 0 for all x an element of [a,b]. prove that if the integral between the boundaries b and a of f(x) dx =0 then f(x) =0 for all x an element of [a,b] (b)Prove that the integral between infinity and 0 of e^-st .sinat dt = a/s^2 + a^2

Complex integrals

(1) let f:C----R be an analytic function such that f(1)=1. Find the value of f(3) (2) Evaluate the integral over & of dz/ z^2 -1 where & is the circle |z-i|=2 (3)Evaluate the integral over & of (z-1/z) dz where & is the line path from 1 to i (4) Evaluate the integral between 2pi and 0 of e^-i@ . e ^e^i@ d@