Evaluate the integral. f10 (4√u + 1)2 du
> 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 the integral. f12 (x – 1)3/x2 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. 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?
> Suppose that you push a book across a 6-meter-long table by exerting a force f (x) at each point from x = 0 to x = 6. What does f60f (x) dx represent? If is measured in newtons, what are the units for the integral?
> (a). Explain the meaning of the indefinite integral ff (x) dx. (b). What is the connection between the definite integral fbaf (x)dx and the indefinite integral ff (x) dx?
> An arc PQ of a circle subtends a central angle θ as in the figure. Let A (θ) be the area between the chord PQ and the arc PQ. Let B (θ) be the area between the tangent lines PR, QR and the arc. Find A(0) lim 0
> (a). How is the length of a curve defined? (b). Write an expression for the length of a smooth curve with parametric equations x = f (t), y = g (t), a < t < b. (c). How does the expression in part (b) simplify if the curve is described by giving y in ter
> If r (t) is the rate at which water flows into a reservoir, what does ft2t1r (t) dt represent?
> (a). Write the definition of the definite integral of a continuous function from a to b. (b). What is the geometric interpretation of fba f (x) dx if f (x) > 0? (c). What is the geometric interpretation of fba f (x) dx if f (x) takes on both positive and
> Determine a region whose area is equal to the given limit. Do not evaluate the limit. lim A" -1 in tan 4n 4n
> Suppose that Sue runs faster than Kathy throughout a 1500-meter race. What is the physical meaning of the area between their velocity curves for the first minute of the race?
> State the rules for approximating the definite integral fbaf (x) dx with the Midpoint Rule, the Trapezoidal Rule, and Simpson’s Rule. Which would you expect to give the best estimate? How do you approximate the error for each rule?
> Explain exactly what is meant by the statement that “differentiation and integration are inverse processes.”
> (a). Given an initial approximation x1 to a root of the equation f (x) = 0, explain geometrically, with a diagram, how the second approximation x2 in Newton’s method is obtained. (b). Write an expression for x2 in terms of x1, f (x1), and f(x1). (c). Wri
> What is a normal distribution? What is the significance of the standard deviation?
> Suppose f (x) is the probability density function for the weight of a female college student, where is measured in pounds. (a). What is the meaning of the integral f1300f (x) dx? (b). Write an expression for the mean of this density function. (c). How ca
> If a rectangle has its base on the -axis and two vertices on the curve y = e-x2, show that the rectangle has the largest possible area when the two vertices are at the points of inflection of the curve.
> (a). What is an antiderivative of a function f? (b). Suppose F1 and F2 are both antiderivatives of f on an interval I. How are F1 and F2 related?
> (a). What is the cardiac output of the heart? (b). Explain how the cardiac output can be measured by the dye dilution method.
> Given a demand function p (x), explain what is meant by the consumer surplus when the amount of a commodity currently available is X and the current selling price is p. Illustrate with a sketch.
> A company modeled the demand curve for its product (in dollars) by the equation Use a graph to estimate the sales level when the selling price is $16. Then find (approximately) the consumer surplus for this sales level. 800,000e/S000 p = x + 20,00
