Mth 74 - Review Problems for Test Test covers Sections 6.-6.5, 7. and 7. For Test # study these problems, the examples in your notes, and the homework.. The base of a solid is the region inside the circle x + y = 4. Find the volume of the solid if every cross section perpendicular to the x-axis is a square with one side in the xy-plane.. Find the average value for f(x) = 3x on the interval [-4, -]. At what x-value(s) does the function assume its average value? 3. The function T (t) = 3 + t t approximates the temperature at t hours past noon on a typical January day in Blacksburg, VA. Find the average temperature between noon and 6: p.m.. Apply the MVT to determine the time when the average temperature is reached. 4. Find the area of the graph bounded by y = x and the x-axis on the interval [, 9]. 5. Find the area of the region bounded by the curve y = e x and the line through the points (,) and (, /e). 6. Consider the region bounded by y = x + 4 and x = y. (a) Use integration with respect to y to find the area of the given region. 7. A region R is bounded by the curve y = x x and the x-axis. (a) Set up the integral to find the area of R. (b) Set up the integral to find the volume of the solid generated by revolving R about the x-axis. (c) Set up the integral to find the volume of the solid generated by revolving R about the y-axis. (d) Set up the integral to find the volume of the solid generated by revolving R about the line x =. 8. A region R is bounded by the curve x = y y and the y-axis. (a) Set up the integral to find the volume of the solid generated by revolving R about the x-axis. (b) Set up the integral to find the volume of the solid generated by revolving R about the y-axis. (c) Set up the integral to find the volume of the solid generated by revolving R about the line x = 4. (d) Set up the integral to find the volume of the solid generated by revolving R about the line y = 3. 9. Find the volume of the solid that is formed by revolving the region bounded by the graph of y = x, y =, x =, and x = about the y-axis.. Find the volume of the solid generated by revolving the region in the first quadrant bounded by y = e x, y = e x, and x = ln 3 about the x-axis.
. The natural length of a certain spring is 6 inches, and a force of 8 lbs is required to keep it stretched 4 inches. Find the work done in each of the following cases: (a) Stretching the spring from a length of 8 inches to a length of 4 inches. (b) Compressing the spring from its natural length to a length of 4 inches.. An upright cylindrical tank is feet in diameter and feet high. If the water in the tank is 8 feet deep, how much work is done in pumping all the water over the edge of the top of the tank? 3. An open tank has the shape of a right circular cone. The vertex of the cone is pointing downwards. The tank is filled with water to a depth of 3/4 its height. If the height of the tank is feet and the radius of the top is 4 feet, find the work done in pumping the water to a height feet above the top of the tank. (Assume water weighs 6.5 lbs/ft 3 ) 4. A piano weighing 45 lbs. is supported by a ft. cable weighing lbs/ft (a) Find the work required to lift the piano 6 ft. by winding the cable on a winch. (b) Find the work required to lift the piano all the way to the top. 5. The vertical end of a tank is a trapezoid. The base that lies on the ground is feet long and the base that forms the top of the tank is 6 feet long (the graph below shows the vertical end of the tank). The tank is 4 feet hight and feet long and is full of water. Find the work done in pumping the water to a height 5 feet above the top of the tank. 4 3 3 6. A tank is full of water. Find the work required to pump the water out of the outlet. Use the fact that water weighs Kg/m 3 m { r=.5 m 6 m
7. Evaluate each of the following: (a) 4e x + e x (b) tan ( + e x(ln x) x (c) ) e x (d) (g) /6 e 3/x x (e) 9x (h) sin (y)e cos (y) dy sin (x) cos 4 x (f) (i) 4x e x + x cos (x) (j) x + x + 4x 3 (k) x 3 ln x (l) x 3 4 x 4 (m) x 4 x 4 (n) (ln x) x (o) tan x sec 4 x (p) π/ sin x cos x + (q) π/ sin x cos x + (r) + ln x x ln x (s) 3xe 4x (t) cos 3 x (u) cos (x) (v) sin x (w) e t cos t dt 8. Find the volume of the solid generated when the region bounded by f(x) = tan x and the x-axis from x = to x = is revolved around the y-axis.
Mth 74 - Answers for Review Problems for Test. V = 8 3 units3. fave =, c = 7 3. Tave = 4, Average temperature occurs about 3:39 pm. 4. A = 4 units 5. A = 3 e e 6. (a) A = 3 3 7. (a) A = (b) V = π units (c) V = π (d) V = π x x (x x ) 9. Using shells: V = π ln x(x x ). Using washers: V = 3π 9. work = 75, π ft-lbs ( x)(x x ) 8. (a) V = π (b) V = π (c) V = π (d) V = π y(y y ) dy (y y ) dy 4 (4 (y y )) dy (y + 3)(y y ) dy. TBA 3. Work = 5875 π 8 4. (a) work = 784 ft-lbs (b) work =4,6 ft-lbs 5. (a) ( x, ȳ) = ( 9, 6 5 ) (b) ( x, ȳ) = (, 7 5 ) 6. work =, 3 ft-lbs 7. integrals
(a) ln + e x + C (b) ln x + C (c) ln cos ( + e x ) + C (d) 3 (e3/ e 3 ) (e) ecos(y) + C (f) (e e) [ ( ) ] (g) u = 3x sin sin () = π 8 3 (h) sin(x) = sin x cos x then u = cos x 3 cos6 x + C (i) Parts x sin(x) + x cos(x) 4 sin(x) + C (j) u = x + 4x 3 ln x + 4x 3 + C (k) Parts x 4 ln x 4 x4 6 + C (l) u = 4 x 4 4 x4 + C (m) u = x sin ( ) x + C (n) u = ln x 3 (ln x)3 + C (o) u = tan x 5 tan5 x + 3 tan3 x + C (p) u = cos x + ln (q) u = cos x tan tan = π 4 (r) u = ln x ln ln x + ln x + C (s) Parts 3 4 xe 4x 3 6 e 4x + C (t) u = sin x sin x 3 sin3 x + C (u) Trig identity sin θ = ( cos(θ)) cos x + C (v) Parts u = sin x x sin x + x + C (w) Parts e t (sin t cos t) + C 8. Parts u = tan x V = π x tan x = π π