Evaluate the integral. f12 (x – 1)3/x2 dx
> Find the average value of the function f (x) = x2 √1 + x3 on the interval [0, 2].
> The demand function for a commodity is given by p = 200 – 0.1x – 0.01x2 Find the consumer surplus when the sales level is 100.
> Find the centroid of the region shown. yA (3, 2)
> A gate in an irrigation canal is constructed in the form of a trapezoid 3 ft wide at the bottom, 5 ft wide at the top, and 2 ft high. It is placed vertically in the canal so that the water just covers the gate. Find the hydrostatic force on one side of t
> Evaluate the integral. f03π/2 |sin x| dx
> A trough is filled with water and its vertical ends have the shape of the parabolic region in the figure. Find the hydrostatic force on one end of the trough. 8 ft 4 ft
> Find the area of the region bounded by the given curves. y = 1 - 2x², y = |x|
> Evaluate the integral. f dt/t2 + 6t+ 8
> Evaluate the integral. f sin x cos (cos x) dx
> For what values of m do the line y = mx and the curve y = x/ (x2 + 1) enclose a region? Find the area of the region.
> 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?
> Find the length of the curve y = fx1√√t, - 1, dt, 1 < x < 16.
> Find the length of the curve y = 1/6 (x2 + 4)3/2, 0 < x < 3.
> Use Simpson’s Rule with n = 10 to estimate the length of the arc of the curve y = 1/x? from (1, 1) to (2, 4).
> Find the length of the curve with parametric equations x= 3t2, y = 2t3, 0 < t < 2.
> Evaluate the integral. f02 (x4 – 3/4x2 + 2/3x -1) dx
> Evaluate the integral. f21 x3 lnx dx
> The height of a monument is 20 m. A horizontal cross section at a distance meters from the top is an equilateral triangle with side 1/4x 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 -axis are squares with one side lying along the base.
> Find the area of the region bounded by the given curves. y = 1/x, y = x², y= 0, x= e
> 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.
> Determine a region whose area is equal to the given limit. Do not evaluate the limit. 10 lim 2 A" i-1 n 2i 5 +
> The “Garbage Project” at the University of Arizona reports that the amount of paper discarded by households per week is normally distributed with mean 9.4 lb and standard deviation 4.2 lb. What percentage of households throw out at least 10 lb of paper a
> Suppose you are asked to estimate the volume of a football. You measure and find that a football is 28 cm long. You use a piece of string and measure the circumference at its widest point to be 53 cm. The circumference 7 cm from each end is 45 cm. Use Si
> Describe the solid whose volume is given by the integral. /2 (a) [* 27 cos?x dx 2т сos?x dx (b) 7[(2 – x²)? – (2 – JI):] dx or
> Let R 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 R (c). The volume generated when R is rotated about the x-axis
> Evaluate the integral. f-12 (x – 2|x|) dx
> Evaluate the integral. f21 (8x3 + 3x2) dx
> Evaluate: (a). f10d/dx (earctan x) dx (b). d/dx f10 earctan x dx (c). d/dx fx0 earctan t dx
> (a). Find the vertical and horizontal asymptotes, if any. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts
> 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 = cos?x, |x|< n/2, y = about x = T/ T/2
> (a). Write f20e3x dx as a limit of Riemann sums, taking the sample points to be right endpoints. Use a computer algebra system to evaluate the sum and to compute the limit. (b). Use the Evaluation Theorem to check your answer to part (a).
> If f60f (x) dx = 10 and f40f (x) dx = 4, find f64f (x) dx.
> Express as a definite integral on the interval [0, π] and then evaluate the integral. lim 2 sin x; Ax
> Evaluate the integral. f 10/ (x – 1) (x2 + 9), dx
> Find the derivative of the function. F (x) = fx0t2/1 + t2, dt
> Graph the function f (x) = cos2x sin3x and use the graph to guess the value of the integral f2π0f (x) dx. Then evaluate the integral to confirm your guess.
> The Fresnel function was defined in Example 4 and graphed in Figures 6 and 7. Figures 6: Figures 7: (a). At what values of does this function have local maxi mum values? (b). On what intervals is the function concave upward? (c). Use a graph to solv
> Let g (x) = f 9x2), where f is twice differentiable for all x, f'(x) > 0 for all x ≠ 0, and f is concave downward on (-∞, 0) and concave upward on (0, ∞). (a). At what numbers does g have an extreme value? (b). Discuss the concavity of g.
> Use a graph to give a rough estimate of the area of the region that lies under the curve y = x√x, 0 < x < 4. Then find the exact area.
> Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). dx Vx? + 1
> Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). cos x V1 + sin x Sin X
> Evaluate the integral. f10 ex/1 + e2x, dx
> Evaluate the integral. f sec θ tan θ/1 + sec θ, dθ
> 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 = Vĩ, y = x; about y = 2
> Evaluate the integral. f tan-1 x dx
> Evaluate the integral. f e3√x dx
> (a). Find the Riemann sum for f (x) = sin x, 0 < x < 3π/2, with six terms, taking the sample points to be right endpoints. (Give your answer correct to six decimal places.) Explain what the Riemann sum represents with the aid of a sketch. (b). Repeat par
> Evaluate the integral. f x/√1 – x4, dx
> Evaluate f10 (x + √1 – x2) dx by interpreting it in terms of areas.
> Evaluate the integral. f41 x3/2 lnx dx
> Evaluate the integral. f41 dt/ (2t + 1)3
> Evaluate the integral. fπ/4-π/4 t4 tan t/2 + cos t, dt
> Evaluate the integral. f50 ye-0.6y dy
> Evaluate the integral. f50 x/x + 10, dx
> Evaluate the integral. f x + 2/√x2 + 4x, dx
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x² – y? = a², x = a + h (where a > 0, h> 0); about the y-axis
> Evaluate the integral. f21 ½ - 3x, dx
> A stone is dropped from the upper observation deck (the Space Deck) of the CN Tower, 450m above the ground. (a). Find the distance of the stone above ground level at time t. (b). How long does it take the stone to reach the ground? (c). With what velocit
> Evaluate the integral. f0π/4 1 + cos2θ/cos2 θ dθ
> (a). Evaluate the Riemann sum for f (x) = x2 – x, 0 (b). Use the definition of a definite integral (with right end - points) to calculate the value of the integral (c). Use the Evaluation Theorem to check your answer to part (b). (d).
> Evaluate the integral. f10 eπt dt
> Evaluate the integral. f10 sin (3πt) dt
> Evaluate the integral. f10v2 cos (v3) dv
> Evaluate the integral. f csc2x/ 1 + cot x, dx
> Evaluate the integral. f10 x/x2 + 1, dx
> Evaluate the integral. f10 (4√u + 1)2 du
> Evaluate the integral. f (1 – x/x)2 dx
> Evaluate the integral. f10 (1 – x)9 dx
> 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
> Evaluate the integral. f0π/3 sin θ + sin θ tan2 θ /sec2θ, dθ
> (a). If g (x) = 1/(√x -1), use your calculator or computer to make a table of approximate values of f12g (x) dx for t = 5, 10, 100, 1000, and 10,000. Does it appear that f∞2g (x) dx is convergent or divergent? (b). Use the Comparison Theorem with f (x) =
> Evaluate the integral. f10 (1 – x9) dx
> Evaluate the integral. fr0 (x4 – 8x + 7) dx
> Use the given graph of f to find the Riemann sum with six subintervals. Take the sample points to be (a) left endpoints and (b) midpoints. In each case draw a diagram and explain what the Riemann sum represents. y= f(x) 6 2. 2.
> (a). Find the vertical and horizontal asymptotes, if any. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts
> (a). Find the vertical and horizontal asymptotes, if any. (b). Find the intervals of increase or decrease. (c). Find the local maximum and minimum values. (d). Find the intervals of concavity and the inflection points. (e). Use the information from parts
> A light is to be placed atop a pole of height feet to illuminate a busy traffic circle, which has a radius of 40 ft. The intensity of illumination I at any point P on the circle is directly proportional to the cosine of the angle (see the figure) and inv
> A canister is dropped from a helicopter 500 m above the ground. Its parachute does not open, but the canister has been designed to withstand an impact velocity of 100 m/s. Will it burst?
> Sketch the graph of a continuous, even function f such that f (0) =, f'(x) = 2x if 0 < x < 1, f'(x) = -1 if 1 < x < 3, and f'(x) = 1 if x > 3.
> Find the local and absolute extreme values of the function on the given interval. f(x) = (In x)/x², [1, 3]
> Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. If f'(c) = 0, then f has a local maximum or minimum at c. 2. If f has an absolute minimum value
> A particle is moving with the given data. Find the position of the particle. a (t) = sin t + 3 cos t, s (0) = 0 v (0) = 2
> Show that if f is a polynomial of degree 3 or lower, then Simpson’s Rule gives the exact value of fba f (x) dx.
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x = 0, x= 9 – y?; about x = -1
> A particle is moving with the given data. Find the position of the particle. v (t) = 2t – 1/ (1 + t2), s (0) = 1
> Find f (x). f"(x) = 2x3 + 3x2 - 4x + 5 f (0) = 2, f(1) = 0
> Find f (x). f"(x) = 1 - 6x + 48x2 f (0) = 1 f'(0) = 2
> Find f (x). f' (u) = u2 + √u /u, f (1) = 3
> Find f (x). f'(t) = 2t - 3 sin t, f (0) = 5
> Use Newton’s method to find all roots of the equation sin x = x2 – 3x + 1 correct to six decimal places.
> Find the local and absolute extreme values of the function on the given interval. f(x) = x + sin 2x, [0, 7]
> 1. If f and g are continuous on [a, b], then 2. If f and g are continuous on [a, b], then 3. If f is continuous on [a, b], then 4. If f is continuous on [a, b], then 5. If f is continuous on [a, b] and f (x) > 0, then 7. If f and g are cont
> A metal storage tank with volume V is to be constructed in the shape of a right circular cylinder surmounted by a hemisphere. What dimensions will require the least amount of metal?
> The velocity of a wave of length L in deep water is where K and C are known positive constants. What is the length of the wave that gives the minimum velocity? L v = K- C C L
> An observer stands at a point P, one unit away from a track. Two runners start at the point S in the figure and run along the track. One runner runs three times as fast as the other. Find the maximum value of the observer’s angle of sig
> Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. x = 1+ y', y = x – 3; about the y-axis
> Find the area of the region bounded by the given curves. y = x², y= 4x – x²
> (a). What is the physical significance of the center of mass of a thin plate? (b). If the plate lies between y = f (x) and y = 0, where a < x < b, write expressions for the coordinates of the center of mass.
> If you have a graphing calculator or computer, why do you need calculus to graph a function?
