Find ds/dθ for the curves : (a) r2 = a2cos2θ (b) rn = ancosnθ

Tangent and Normal (II) (Differential Calculus) Find ds/dθ for the curves : (a) r2 = a2cos2θ (b) rn = ancosnθ See attached file for full problem description.

Find ds/dr for the curves: (a) r = aθ (b) r = a/θ

Tangent and Normal (III) (Differential Calculus) Find ds/dr for the curves: (a) r = aθ (b) r = a/θ See attached file for full problem description. ...continues

1. Prove that if a and b are odd integers then a2 – b2 is divisible by 8. 2. Prove that if a is an odd integer , then {a2 + (a + 2)2 + (a + 4)2 + 1} is divisible by 12.

Number Theory The Problems of the Euclid’s Division Lemma 1. Prove that if a and b are odd integers then a2 – b2 is divisible by 8. 2. Prove that if a is an odd integer , then {a2 + (a + 2)2 + (a ...continues

Double Integrals

Please see the attached file for full problem description. --- Find the volume of the region that lies under the graph of the paraboloid z = x^2 + y^2 + 2 and over the rectangle R = {(x, y) | -1 and in two ways (a) by using Cavalieri's principle to write the volume as an iterated integral that results from slicing ...continues

Find the center of mass of the region

Please see the attached file for full problem description. --- Here is the problem: Find the center of mass of the region bounded by the parabola y = 8 -2x^2 and the x-axis a) if the density lambda is constant and b) if the density lambda = 3y

Find the volume under the plane...

Here is the problem: Find the volume under the plane z = 4x + 2y + 25 and over the region bounded by y = x^2 - 10 and y = 31 - (x-1)^2

Function problem

If F(x)=x/x+1 find F(x+h)-F(x)/h where h is not 0. This problem wants you to substitute F(x+h)-F(x)/h into F(x)=x/x+1.

Shell Method

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y axis. 1. y=x. Line goes from (0,0) to to (2,2). Picture shows a thin yellow rectangle going from about x=1.5 vertical to the y=x line. 2) y=sq. root of x. Line goes from (0,0) t ...continues

Center of Mass

Find the center of mass of a plate that is shaped like the region between y=x^2 and y=2x, where the density varies as 1 + x + y. (Question is also included in attachment)

Integral

Evaluate the attached integral Write an equivalent iterated integral with the order of integration reversed. Evaluate this new integral and check that your answer agrees with part (a) (Question is included in attachment) Please sent response as attachment.