Water in an open bowl evaporates at a rate proportional to the area of the surface of the water. (This means that the rate of decrease of the volume is proportional to the area of the surface.) Show that the depth of the water decreases at a constant rate, regardless of the shape of the bowl.
> Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. у — Зx? — 2х, у—х3 — Зх + 4
> Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. y = (x² + 1)²' y = x' – x, x>0
> Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. у —х sin(x"), у — х", x> 0 y y
> Evaluate the integral and interpret it as the area of a region. Sketch the region. L,13* – 2*| dx
> Evaluate the integral and interpret it as the area of a region. Sketch the region. 1/2 " | sin x – cos 2x| dx
> Use calculus to find the area of the triangle with the given vertices. (2, 0), (0,2), (-1, 1)
> Use calculus to find the area of the triangle with the given vertices. (0, 0), (3, 1), (1, 2)
> Sketch the region enclosed by the given curves and find its area. In x y = - (In x)² y =
> Sketch the region enclosed by the given curves and find its area. x² y 1+ x? y 1 + x³
> Find the area of the shaded region. 4. yA x= y? – 4y (-3, 3) x= 2y – y²
> Find the area of the shaded region. y AI y=r (1, 1) x= 8 y = 1/x
> Use the graph of
> Use your computer algebra system to differentiate
> Use a graph of
> Show that where And a² + b² – 625 0 = arccoS 2ab a? = (9 + x cos a)² + (31 – x sin a)? b? = (9 + x cos a)² + (x sin a – 6)²
> power model often provides a more accurate fit than a quadratic model for a Lorenz function. If you have a computer with Maple or Mathematica, fit a power function (y – axk) to the data in Problem 2 and use it to estimate the Gini index
> The following table gives values for the Lorenz function in the years 1970, 1980, 1990, and 2000. Use the method of Problem 2 to estimate the Gini index for the United States for those years and compare with your answer to Problem 2(c). Do you notice a t
> The following table (derived from data supplied by the US Census Bureau) shows values of the Lorenz function for income distribution in the United States for the year 2010. (a) What percentage of the total US income was received by the richest 20% of t
> (a) Show that the Gini index G is twice the area between the Lorenz curve and the line y = x, that is, (b) What is the value of G for a perfectly egalitarian society (everybody has the same income)? What is the value of G for a perfectly totalitarian soc
> Find the average value of the function on the given interval. f(t) = esin cos t, [0, 7/2]
> Sketch the region enclosed by the given curves and find its area. x> 0 y /1 + x y 9 - x
> Find the average value of the function on the given interval. g(t) [1, 3] /3 + t²
> A variable force of 5x22 pounds moves an object along a straight line when it is x feet from the origin. Calculate the work done in moving the object from x = 1 ft to x = 10 ft.
> How much work is done when a hoist lifts a 200-kg rock to a height of 3 m?
> A 360-lb gorilla climbs a tree to a height of 20 ft. Find the work done if the gorilla reaches that height in (a) 10 seconds (b) 5 seconds
> If f is a continuous function, what is the limit as h → 0 of the average value off on the interval [x, x + h]?
> Find the average value of the function f(t) − sec2t on the interval [0,
> A force of 30 N is required to maintain a spring stretched from its natural length of 12 cm to a length of 15 cm. How much work is done in stretching the spring from 12 cm to 20 cm?
> Each integral represents the volume of a solid. Describe the solid. 2пх сos x dx
> Each integral represents the volume of a solid. Describe the solid. 1/2 2T cos'x dx Jo
> The linear density in a rod 8 m long is / where x is measured in meters from one end of the rod. Find the average density of the rod.
> The graphs of two functions are shown with the areas of the regions between the curves indicated. (a) What is the total area between the curves for 0 < x< 5? (b) What is the value of f [f(x) – g(x)]dx? y 27 12 4 3. 2.
> Find the average value of the function on the given interval. S(к) — х?/(х3 + 3)?, [-1, 1]
> Find the average value of the function on the given interval. д(х) — 3 сos x, [-п/2, п/2]
> Find the average value of the function on the given interval. f(x) = /x, [0, 4]
> Find the average value of the function on the given interval. f(x) — Зх? + 8x, [-1,2] –1, 2]
> The table shows values of a force function f(x) where x is measured in meters and f(x) in newtons. Use the Midpoint Rule to estimate the work done by the force in moving an object from x = 4 to x = 20. 68 10 1 4 12| 14 16 18 f(x) 5.8 7.0 | 8.8 | 9.6
> The figure shows a horizontal line y = c intersecting the curve y = 8x - 27x3. Find the number c such that the areas of the shaded regions are equal.
> There is a line through the origin that divides the region bounded by the parabola y = x - x2 and the x-axis into two regions with equal area. What is the slope of that line?
> (a) Find a positive continuous function f such that the area under the graph off from 0 to t is A(t) = t3 for all t. 0. (b) A solid is generated by rotating about the x-axis the region under the curve y – f(x), where f is a positive function and x > 0.
> If the tangent at a point P on the curve y = x3 intersects the curve again at Q, let A be the area of the region bounded by the curve and the line segment PQ. Let B be the area of the region defined in the same way starting with Q instead of P. What is t
> Suppose we are planning to make a taco from a round tortilla with diameter 8 inches by bending the tortilla so that it is shaped as if it is partially wrapped around a circular cylinder. We will fill the tortilla to the edge (but no more) with meat, chee
> Sketch the region enclosed by the given curves and find its area. у — x, у— 2x, х+у—3, х>0
> Suppose the graph of a cubic polynomial intersects the parabola y = x2 when x = 0, x = a, and x = b, where 0 < a < b. If the two regions between the curves have the same area, how is b related to a?
> A cylindrical container of radius r and height L is partially filled with a liquid whose volume is V. If the container is rotated about its axis of symmetry with constant angular speed
> A clepsydra, or water clock, is a glass container with a small hole in the bottom through which water can flow. The “clock” is calibrated for measuring time by placing markings on the container corresponding to water l
> A paper drinking cup filled with water has the shape of a cone with height h and semi- vertical angle
> The figure shows a curve C with the property that, for every point P on the middle curve y = -x2, the areas A and B are equal. Find an equation for C. yA y= 2x? C y=x² A
> A sphere of radius 1 overlaps a smaller sphere of radius r in such a way that their intersection is a circle of radius r. (In other words, they intersect in a great circle of the small sphere.) Find r so that the volume inside the small sphere and outsid
> Archimedes’ Principle states that the buoyant force on an object partially or fully submerged in a fluid is equal to the weight of the fluid that the object displaces. Thus, for an object of density 0 floating partly submerged in a flui
> (a) Show that the volume of a segment of height h of a sphere of radius r is (b) Show that if a sphere of radius 1 is sliced by a plane at a distance x from the center in such a way that the volume of one segment is twice the volume of the other, then x
> A cylindrical glass of radius r and height L is filled with water and then tilted until the water remaining in the glass exactly covers its base. (a) Determine a way to “slice” the water into parallel rectangular cros
> Sketch the region enclosed by the given curves and find its area. у — 1/х, у— х, у—\x, х>0
> Find the area of the region bounded by the given curves. y = x', y = 4x – x²
> Let R1 be the region bounded by y = x2 , y = 0, and (a) Is there a value of b such that /1 and /2 have the same area? (b) Is there a value of b such that /1 sweeps out the same volume when rotated about the x-axis and the y-axis? (c) Is there a value of
> (a) Find the average value of the function f(x) – 1/sx on the interval [1, 4]. (b) Find the value c guaranteed by the Mean Value Theorem for Integrals such that fave = f(c). (c) Sketch the graph of f on [1, 4] and a rectangle whose area is the same as th
> A steel tank has the shape of a circular cylinder oriented vertically with diameter 4 m and height 5 m. The tank is currently filled to a level of 3 m with cooking oil that has a density of 920 kg/m3. Compute the work required to pump the oil out through
> A tank full of water has the shape of a paraboloid of revolution as shown in the figure; that is, its shape is obtained by rotating a parabola about a vertical axis. (a) If its height is 4 ft and the radius at the top is 4 ft, find the work required to p
> A 1600-lb elevator is suspended by a 200-ft cable that weighs 10 lb/ft. How much work is required to raise the elevator from the basement to the third floor, a distance of 30 ft?
> (a) The base of a solid is a square with vertices located at (1, 0), (0, 1), (21, 0), and (0, 21). Each cross-section perpendicular to the x-axis is a semicircle. Find the volume of the solid. (b) Show that by cutting the solid of part (a), we can rearra
> The height of a monument is 20 m. A horizontal cross section at a distance x meters from the top is an equilateral triangle with / meters. Find the volume of the monument.
> The base of a solid is the region bounded by the parabolas y = x2 and y = 2 - x2. Find the volume of the solid if the cross-sections perpendicular to the x-axis are squares with one side lying along the base.
> The base of a solid is a circular disk with radius 3. Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base.
> Sketch the region enclosed by the given curves and find its area. у 3 sinhx, у —е , х—0, х—2
> Each integral represents the volume of a solid. Describe the solid. 2т (6 — у)(4у — у?) dy
> Each integral represents the volume of a solid. Describe the solid. S T (2 – sin x)² dx
> Let / be the region bounded by the curves y = 1 - x2 and y = x6 - x + 1. Estimate the following quantities. (a) The x-coordinates of the points of intersection of the curves (b) The area of / (c) The volume generated when/is rotated about the x-axis (d)
> Let / be the region bounded by the curves y = tan(x2), x = 1, and y = 0. Use the Midpoint Rule with n = 4 to estimate the following quantities. (a) The area of/ b) The volume obtained by rotating / about the x-axis
> Let / be the region in the first quadrant bounded by the curves y = x3 and y = 2x 2 x2. Calculate the following quantities. (a) The area of / (b) The volume obtained by rotating / about the x-axis (c) The volume obtained by rotating / about the y-axis
> Find the volumes of the solids obtained by rotating the region bounded by the curves y = x and y = x2 about the following lines. (a) The x-axis (b) The y-axis (c) y = 2
> Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. - y = Vx, y = x²; about y = 2
> Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. у — соs х, [|x| < п/2, у — %: about.х — п, п/2
> Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. у %3 tan x, у — х, х — п/3; about the y-axis TT
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x? – y? = a², x = a + h (where a>0, h > 0); about the y-axis
> Sketch the region enclosed by the given curves and find its area. y = x*, y = 2 - |x| у — х",
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = x? + 1, y = 9 – x²; about y = -1
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x = 0, x = 9 – y²; about x = -1
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x = 1 + y², y - x – 3; about the y-axis
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. _y = 2x, y = x²; about the x-axis %3D
> Find the area of the region bounded by the given curves. у 3 Vx, у— х?, х—2 y
> Find the area of the region bounded by the given curves. у у — х? — 2х sin(тx/2), 2.x y
> Find the area of the region bounded by the given curves. x + y = 0, x = y² + 3y
> Find the area of the region bounded by the given curves. у —1- 2x?, у — \x| y y = |x|
> Find the area of the region bounded by the given curves. у — x, у— —Vx, у—х— 2
> (a) An outfielder fields a baseball 280 ft away from home plate and throws it directly to the catcher with an initial velocity of 100 ft/s. Assume that the velocity v(t) of the ball after t seconds satisfies the differential equation dv/dt − 2 1 10v beca
> Sketch the region enclosed by the given curves and find its area. y = cos x, y=1 – cos x, 0<x<T
> In this problem we calculate the work required for a pitcher to throw a 90-miyh fastball by first considering kinetic energy. Suppose an object of mass m, moving in a straight line, is acted on by a force F = F(s) that depends on its position s. Accordin
> It may surprise you to learn that the collision of baseball and bat lasts only about a thousandth of a second. Here we calculate the average force on the bat during this collision by first computing the change in the ball’s momentum. Th
> Find the average value of the function on the given interval. h(u) — (In u)/и, [1,5]
> Find the average value of the function on the given interval. h(x) cos*x sin x, [0, 7]
> If fave [a, b] denotes the average value of f on the interval [a, b] and a b — с Save[c, b] b — а c - a fave[a, b] fave [a, c] + b — а
> Prove the Mean Value Theorem for Integrals by applying the Mean Value Theorem for derivatives (see Section 4.2) to the function /
> Use the diagram to show that if f is concave upward on [a, b], then a + b fave >f 2 y a +b b a 2.
> Use the result of Exercise 5.5.83 to compute the average volume of inhaled air in the lungs in one respiratory cycle.
> If a freely falling body starts from rest, then its displacement is given by / 2tt2. Let the velocity after a time T be vT. Show that if we compute the average of the velocities with respect to t we g / 2vT, but if we compute the average of the velocit
> In Example 3.8.1 we modeled the world population in the second half of the 20th century by the equation P(t) = 2560e0.017185t. Use this equation to estimate the average world population during this time period.
> Sketch the region enclosed by the given curves and find its area. y = 2x, y=x², 0<x<6
> (a) A cup of coffee has temperature 95°C and takes 30 minutes to cool to 61°C in a room with temperature 20°C. Use Newton’s Law of Cooling (Section 3.8) to show that the temperature of the coffee after t
> The velocity v of blood that flows in a blood vessel with radius R and length l at a distance r from the central axis is where P is the pressure difference between the ends of the vessel and is the viscosity of the blood (see Example 3.7.7). Find the av
> In a certain city the temperature (in °F) t hours after 9 am was modeled by the function Find the average temperature during the period from 9 am to 9 pm. T(1) = 50 + 14 sin 12
