Questions from General Calculus

Q: A region R is shown. Decide whether to use polar coordinates

A region R is shown. Decide whether to use polar coordinates or rectangular coordinates and write∬R f (x, y) dA as an iterated integral, where f is an arbitrary continuous function on...

Q: Evaluate the iterated integral. ∫_0^1 ∫_0

Evaluate the iterated integral. ∫_0^1 ∫_0^(e^y) √(1+e^y) dw dv

Q: Evaluate the given integral by changing to polar coordinates. ∬

Evaluate the given integral by changing to polar coordinates. ∬D x2y dA, where D is the top half of the disk with center the origin and radius 5.

Q: Evaluate the given integral by changing to polar coordinates. ∬

Evaluate the given integral by changing to polar coordinates. ∬R (2x – y) dA, where R is the region in the first quadrant enclosed by the circle x2 + y2 = 4 and the lines x = 0 and y = x

Q: Evaluate the given integral by changing to polar coordinates. ∬

Evaluate the given integral by changing to polar coordinates. ∬R sin (x2 + y2) dA, where R is the region in the first quadrant between the circles with center the origin and radii 1 and 3.

Q: Find the mass and center of mass of the lamina that occupies

Find the mass and center of mass of the lamina that occupies the region D and has the given density function ρ. D is enclosed by the curves y = 0 and y = cos x, -π/2 < x < π/2; ρ (x, y) = y

Q: A lamina occupies the part of the disk x2 + y2 <

A lamina occupies the part of the disk x2 + y2 < 1 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis.

Q: The plane 4x - 3y + 8z = 5 intersects the cone

The plane 4x - 3y + 8z = 5 intersects the cone z2 = x2 + y2 in an ellipse. (a). Graph the cone and the plane, and observe the elliptical intersection. (b). Use Lagrange multipliers to find the highest...

Q: The boundary of a lamina consists of the semicircles y = √(

The boundary of a lamina consists of the semicircles y = √(1 - x^2 ) and y = √(4 - x^2 ) together with the portions of the x-axis that join them. Find the center of mass of the lamina if the density a...

Q: Find the center of mass of the lamina in Exercise 13 if

Find the center of mass of the lamina in Exercise 13 if the density at any point is inversely proportional to its distance from the origin. Exercise 13: The boundary of a lamina consists of the semi...