Ch.8 1. When the region R={(x, y) x 21,0 5 y 51/x is rotated around the...
1. In this problem, we will investigate the Paradox of Gabriel's Horn: There exists a hollow solid that can be completely filled with a finite amount of paint, and yet no amount of paint will be sufficient to cover its entire surface. Let S be the surface (Gabriel's Horn) obtained by rotating the curve yaround the r-axis for 2 1, and let Sb be the surface obtained by rotating the curve y- around the r-axis for 1 3Sb. IfV and...
problem 3 pls
Problem 3. Consider the curve {> 1, y = 1/x}. Compute the length of the part of this curve lying to the left of the line x = a for any a > 1. Show that the length of the whole curve is thus infinite. Compute the area of the surface obtained by rotating this curve about about the x axis by computing the corresponding improper integral; it should be infinite. What is the area of the...
Find the area of the surface given by z = f(x, y) over the region R. (Hint: Some of the integrals are simpler in polar coordinates.) f(x, y) = x2 + y2, R = {(x, y): 0 = f(x,y) 3}
Let R be the region in the first quadrant bounded by the x-axis and the graphs of y = in(x) and y=5-x, as shown in the figure above. a) Find the area of R. b) Region R is the base of a solid. For the solid, each cross-section perpendicular to the x-axis is a right isosceles triangle whose leg falls in the region. Write, but do not evaluate, an expression involving one or more integrals that gives the volume of the solid. c)...
8. Find the area of the surface given by z - f(x, y) over the region R. f(x,y)- 42-x2-y2, R = {(x,y): x2 +y2 29
8. Find the area of the surface given by z - f(x, y) over the region R. f(x,y)- 42-x2-y2, R = {(x,y): x2 +y2 29
Let R be the region bounded by the graphs of ysx2-3 and x y2 (a) Find the area of R. (Round your answer to four decimal places.) b) Find the volume of the solid d generated when R is rotated about the vertical line 4. (Round your answer to four decimal places) (c) Write, but do not evaluate, an expression involving one or more integrals to find the volume of the solid generated when R is rotated about the your...
Instructions: Show all your work for FULL credit. Calculators are NOL final answer. Neatness is highly appreciated. 1. A region R, bounded by y 2x, y 6-x, and x-axis, is rotated around the y-axis. Sketch the region R, in the box a) 15 strip/slice you will use to find the volume of the solid of revolution. b) Write the definite integral that gives a X the volume of the solid of revolution. (DO NOT evaluate the integral.) Find the circumference...
3. Find the total area of the region enclosed by the curve y=xsin x and the x-axis from (0,37). The graph is shown below. Show all of your work! (Hint: You will need multiple integrals here)! (8 pts)
5 pts) Consider the region bounded by the curves y 9, y and r 1 r-+64 If this region is revolved around the x - axis, the volume of the resulting solid can be computed in (at least) two different ways using integrals. (Sketching the graph of the situation m (a) First of all it can be computed as a single integral h(r)dr where o and This method is commonly called the method of Enter 'DW' for Disks/Washers or 'CS...
(2) The area of the surface with equation z = f(x,y). (x,y) E D. where fra f, are continuous, is A(S) = SVGC3. y)]? + [f;(x, y)]? +T dA If you attempt to use Formula 2 to find the area of the top half of the sphere x + y2 + 2? = a, you have a slight problem because the double integral is improper. In fact, the integrand has an infinite discontinuity at every point of the boundary circle...