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Multiple Integrals, Vector Fields, Hemispheres and Divergence Theorem

B10. (a) State the Divergence Theorem, being careful to explain any notation you use and any conditions that must apply. The vector field B is given by B = Rcos θ(cos θR - sin θ ^θ ) in spherical polar coordinates (R; θ; φ). This field exists in a region which includes the hemisphere x2 + y2 + z2


Evaluate e to the power of 3x minus 4 divided by e to the power of x between the ordinate limits -2 and -3.


Integrate 5-3e to the power of4x which is divided by e to the power of x. 4x x trying to write it would be : 5-3e divided by e


Integrate with respect to x (3+4x) to the power -1


Dy/dx = x³ - √x + 3 - secxtanx Find y = I got the following answer:y = x4/4 - 2x (3/2)/3 + 3x - secx Is it correct????

Partial Fractions

Find the integral of a polynomial fraction. See attached file for full problem description.

Simpson's Rule

Use problems 8 and 9 on p. 348 as an outline to write a clear explanation why Simpson's rule is a good way to approximate definite integrals over a finite interval. The questions are attached, I need help explaining each step of the problem, with a few different proofs of how this actually works.


Please solve and explain. Write the expression for the Riemann sum of f(x) = x^2 - 4x on the interval [0,8] with n uniform subintervals using the right hand endpoints of the subintervals. Do not evaluate. Using the Reiman Sum, write the definition of the definite integral 8 to 0 (x^2 - 4x)dx. Do not evaluate. Using

4 Definite Integrals

Evaluate the integrals using the following values (i) For integral 4 on the top, 2 on the bottom x^3 dx = 60 (ii) For the integral 4 on the top, 2 on the bottom x dx = 6 (iii) For the integral 4 on the top, 2 on the bottom dx = 2


Please show how to solve 47 and 48 of the attachment. Please offer as much explanation as possible.


For 21 and 22 on the attached page, please set up a definite integral that gives the area of the region. (Do not evaluate the integral). Please offer as much explanation as possible.

Volume and integration

Find the region enclosed by x=3y and x=-y^2+4. Set up integrals both shell and disc that represent the volume generated when this region is revolved about y=4. Set it up, do not work to completion.


Please explain and solve the following. Use the midpoint rule with n = 4 to approximate the area of the region bounded by the graph of the function and the x axis over the indicated interval. f(x) = x^2 + 3 [0,2]


Please explain how to solve the attached problems (as much explanation as possible) and solve to the specified answers. Find a formula for the sum of n terms. Use the formula to find the limit as n approaches infinity.


Please explain how to do problem 23 on the attached scan. Answer is that the shaded region falls between 12.5 square units and 16.5 square units. As much explanation as possible please. Please also explain how to do 27, 28, 29 and 30 on the attached scan. As much explanation as possible. Please solve the problems. Answer


Please explain how to do the following problem. Find the limit of s(n) as n approaches infinity. s(n) = 1/n^2 [n(n+1)/2]