Question: Determine whether each integral is convergent or
Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
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> Use the Comparison Theorem to determine whether the integral is convergent or divergent. sec'x -dx lo xVx
> Use the Comparison Theorem to determine whether the integral is convergent or divergent. 2 + e -
> Use the Comparison Theorem to determine whether the integral is convergent or divergent. x' + 1 .3
> Show that 1/3 Tn + 3.2Mn = S2n.
> Find the number b such that the line y = b divides the region bounded by the curves y = x2 and y = 4 into two regions with equal area.
> Sketch the region and find its area (if the area is finite). s = {(x, y) | –2 < r< 0, 0 < y< 1//I + 2 }
> A container in the shape of an inverted cone has height 16 cm and radius 5 cm at the top. It is partially filled with a liquid that oozes through the sides at a rate proportional to the area of the container that is in contact with the liquid. (The surfa
> Find the area of the shaded region. YA x= y? - 4y (-3, 3) x= 2y - y?
> Let g (x) = fx0 f (t) dt, where f is the function whose graph is shown. (a). Evaluate g (x) for x = 0, 1, 2, 3, 4, 5 and 6. (b). Estimate g (7). (c). Where does b have a maximum value? Where does it have a minimum value? (d). Sketch a rough graph of g.
> Sketch the region and find its area (if the area is finite). S = {(x, y) | 0 < x < n/2, 0 < y< sec?r}
> Sketch the region and find its area (if the area is finite). S = {(x, y) | x > 0, 0 < y< xe*}
> Sketch the region and find its area (if the area is finite). S = {(x, y) | 0 < y < 2/(x² + 9)}
> Find the area enclosed by the x-axis and the curve x = 1 + et, y = t – t2.
> Find the area enclosed by the curve x = t2 – 2t, y = √t and the y-axis.
> Use the parametric equations of an ellipse, x = a cos θ, y = b sin θ, 0 < θ < 2π, to find the area that it encloses.
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. csc x dx J2/2
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. dx e - 1
> Graph f (x) = sin (ex) and use the graph to estimate the value of t such that ft+1tf (x) dx is a maximum. Then find the exact value of t that maximizes this integral.
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 dy 4y – 1 Je
> Find the area of the shaded region. yA x= y?- 2 y=1 x= e y=-
> A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is a (t) = 60t, at which time the fuel is exhausted and it becomes a freely “falling” body. Fourteen seconds later, the rocket’s parachute opens, and the (d
> If the birth rate of a population is b (t) = 2200e0.024t people per year and the death rate is d (t) = 1460e0.018t people per year, find the area between these curves for 0 < t < 10. What does this area represent?
> A cross-section of an airplane wing is shown. Measurements of the thickness of the wing, in centimeters, at 20-centimeter intervals are 5.8, 20.3, 26.7, 29.0, 29.0, 27.6, 26.3, 28.8, 20.5, 15.1, 8.7 and 2.8. Use Simpson’s Rule to estima
> The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use Simpson’s Rule to estimate the area of the pool. 5.6 5.0 6.8 4.8 4.8 7.2 6.2
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 dx 3— х
> Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race. Use Simpson’s Rule to estimate how much farther Kell
> Sketch the curves y = cos x and y = 1 – cos x, 0 < x < π, and observe that the region between them consists of two separate parts. Find the area of this region.
> Sketch the region that lies between the curves y = cos x and y = sin 2x and between x = 0 and x = π/2. Notice that the region consists of two separate parts. Find the area of this region.
> Show that f10(1 – x2) n dx = 22n(n!)2/ (2n + 1)! Hint: Start by showing that if denotes the integral, then 2k + 2 - Ik 2k + 3 Ik+1
> 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 = x cos x, y=x10
> 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 In x, y = VI – I (х — 1
> 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 = y = x* – x, x 0 (x² + 1)*"
> A high-speed bullet train accelerates and decelerates at the rate of 4ft/s2. Its maximum cruising speed is 90 mi/h. (a). What is the maximum distance the train can travel if it accelerates from rest until it reaches its cruising speed and then runs at th
> Find the area of the shaded region. y. y = Vx+2 x= 2 1 х+1
> 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 = x sin(x²), y = x*
> Sketch the region enclosed by the given curves and find its area. y = 3x², y= 8r², 4x + y = 4, x> 0 %3D
> Sketch the region enclosed by the given curves and find its area. y = 1/x, y= x, y=}x, x>0
> Sketch the region enclosed by the given curves and find its area. у %3D сos x, у — 2 — cos x, 0x<2п
> Sketch the region enclosed by the given curves and find its area. y = e", y = xe", x= 0
> A hemispherical bubble is placed on a spherical bubble of radius 1. A smaller hemispherical bubble is then placed on the first one. This process is continued until chambers, including the sphere, are formed. (The figure shows the case n = 4.) Use mathema
> Sketch the region enclosed by the given curves and find its area. y = x', y = 4x – x?
> Sketch the region enclosed by the given curves and find its area. у — 12 — х', у-х*— 6 y= x² – 6
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. у %3 sin x, у— 2х/т, х>0
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. x= 2y', x= 4 + y? %3D
> To prove Theorem 1, let F and G be any two antiderivatives of f on I and let H = G - F. (a). If x1 and x2 are any two numbers in I with x1 < x2, apply the Mean Value Theorem on the interval [x1, x2] to show that H (x1) = H (x2). Why does this show that H
> Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. 4x + y? = 12, x= y
> Find the area of the shaded region. yA y = 5x – x? ((4, 4) y=x
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. -/2 e dy
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. J. Tr? + 2)²
> In an automobile race along a straight road, car A passed car B twice. Prove that at some time during the race their accelerations were equal.
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 dx V1 +x 4.
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 1 (x – 2)2
> Show that ½ (Tn + Mn) = T2n.
> (a). Graph the functions f (x) = 1/x1.1 and g (x) = 1/x0.9 in the viewing rectangles [0, 10] by [0, 1] and [0, 100] by [0, 1]. (b). Find the areas under the graphs of f and g from x = 1 to x = t and evaluate for t =10, 100, 104, 106, 1010, and 1020. (c).
> Sketch the region and find its area (if the area is finite). s = {(x, y) | x > -2, 0 < y<e?}
> If the units for are feet and the units for a (x) are pounds per foot, what are the units for da/dx? What units does f82 a(x) dx have?
> Sketch the region and find its area (if the area is finite). S = {(r, y) | x < 1, 0 < y< e*}
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. In x dx
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. *2 (2' In z dz Jo
> Find the area under the curve y = 1/x3 from x = 1 to x = t and evaluate it for t = 10, 100, and 1000. Then find the total area under this curve for x > 1.
> Find the minimum value of the area of the region under the curve y = x + 1/x from x = a to x = a + 1.5, for all a > 0.
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 33 " (x – 1)-1/5 dx
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. хр (х — 6)° J6 4.
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. *14 dx -2 x + 2
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 3 dx
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. dx. e2* + 3
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. Je x(In x)
> If x is measured in meters and f (x) is measured in newtons, what are the units for f0100 f (x) dx?
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. In x dx .3
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. x? dx 9 + x*
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. x'e dx
> The line y = mx + b intersects the parabola y = x2 in points A and B. (See the figure.) Find the point P on the arc AOB of the parabola that maximizes the area of the triangle PAB. y y =x B y = mx +b P
> (a). The base of a solid is a square with vertices located at (1, 0), (0, 1) and (0, -1). 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 rearrange it t
> Which of the following integrals are improper? Why? (a) f dx. 2x - 1 (b) Jo 2x 1 dx :- 1 sin x dx -1+x* (d) * In(x – 1) dx
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. In x dx
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. 9. re dr
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. se-* ds -5s Jo
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. * cos at dt
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. x +1 - dx x? + 2x
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. dx
> Since raindrops grow as they fall, their surface area increases and therefore the resistance to their falling increases. A raindrop has an initial downward velocity of 10 m/s and its downward acceleration is If the raindrop is initially 500 m above the
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. xedx
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. L – 3y°) dy
> Use an integral to estimate the sum ∑(i=1)^1000 √i.
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. sin e de
> Determine whether each integral is convergent or divergent. Evaluate those that are convergent. -1 -21 dt
> Use the Table of Integrals on Reference Pages 6–10 to evaluate the integral. | tan'(7x) dx TX
> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) In x -dx, п 3D 10 dx, n= 1 +х
> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) *1/2 sin(r*) dx, n= 4
> Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) 1 + x² dx, n = 8
> A bacteria population starts with 400 bacteria and grows at a rate of r (t) = (450.268) e1.12567t bacteria per hour. How many bacteria will there be after three hours?
> An oil storage tank ruptures at time t = 0 and oil leaks from the tank at a rate of r (t) = 100e-0.01t liters per minute. How much oil leaks out during the first hour?
> A manufacturer of lightbulbs wants to produce bulbs that last about 700 hours but, of course, some bulbs burn out faster than others. Let F (t) be the fraction of the company’s bulbs that burn out before t hours, so F (t) always lies be
> (a). If f (x) = ln x, 1 < x < 4, use the commands discussed in Exercise 9 to find the left and right sums for n = 10, 30, and 50. Exercise 9: Some computer algebra systems have commands that will draw approximating rectangles and evaluate the sums of t
> If P (a, a2) is any point on the parabola y = x2, except for the origin, let Q be the point where the normal line intersects the parabola again. Show that the line segment PQ has the shortest possible length when a = 1/√2. y P
> A model for the basal metabolism rate, in kcal/h, of a young man is R (t) = 85 – 0.18 cos (πt/12), where t is the time in hours measured from 5:00 AM. What is the total basal metabolism of this man, f240 R (t) dt, over a 24-hour time period?
> Which of the following areas are equal? Why? yA y y = 2xe" y y= ein x sin 2.r
> Evaluate f10 x√1 – x4, dx by making a substitution and interpreting the resulting integral in terms of an area.
> Use (a) the Midpoint Rule and (b) Simpson’s Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.) Compare your results to the actual value to determine the error in each approx
> Evaluate f2-2 (x + 3) √4 – x2, dx by writing it as a sum of two integrals and interpreting one of those integrals in terms of an area.
