Calculus Answers

Find the surface area of the part of the surface 2y+ 4z-x^2= 5 that lies above the triangle with vertices (0,0),(2,0), and (2,4).

Find the mass and center of mass of the lamina that occupies the region bounded by y=x+ 2 and y=x^2, where the density at any point is given by σ(x,y) = 3x^2.

How can the surface of a sphere S2 of radius R be defined in terms of spherical coordinates? Express the surface element dS on a sphere S2 in spherical coordinates and
explain its form using words and diagrams. Determine the area of the sphere by evaluating the integral R
S2
dS. Calculate also R
S2
dS and explain why these two surface integrals
are not equal.

Use the Fourier transform method to solve

y'''-3y' = w(x)

on the real line. Write the solution as a Green’s function, while stating the definition of the convolution in this context.

An object is moving so that its speed after t minutes is v (t ) = 1 + 4t + 3t2 meters per minute. How far does the object travel during the 3rd minute?

A wound is healing in such a way that t days since Monday, the area of the wound has been decreasing at a rate of -3(t+2)^-2 sq cm per day. If on Tuesday, the area of the wound was 2 sq cm, what was the area of the wound on Monday?

If f(x,y,z)=(x2y2−2y3z2−4x3z2+xyz) , the gradient of f at (1,−2,−1) is

If f(x,y,z)=(x2y2−2y3z2−4x3z2+xyz) , the gradient of f at (1,−2,−1) is

If u=5t2i+j−t k and v=sinti−costj then, at t=0, ddt(u×v)=

Determine whether the following series converges:
S=1-(1/2)-(1/2^2)+(1/2^3)+(1/2^4)-(1/2^5)-(1/2^6)...
1+(1/3sqrt(2))+(1/3sqrt(3)+...+(1/3sqrt(k))...