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mathematics
calculus

Questions and Answers of Calculus

Use the Table of Integrals on the Reference Pages to evaluate the integral. a. ∫ √(4x2 - 4x -3) dx b. ∫ cosx √(4 + sin2x) dx
Verify Formula 33 in the Table of Integrals (a) By differentiation (b) By using a trigonometric substitution.
Is it possible to find a number n such that
Use (a) The Trapezoidal Rule, (b) The Midpoint Rule, (c) Simpson's Rule with to approximate the given integral. Round your answers to six decimal places
Estimate the errors involved in Exercise 63, parts (a) and (b). How large should be in each case to guarantee an error of less than 0.00001?
The speedometer reading (v) on a car was observed at 1-minute intervals and recorded in the chart. Use Simpson's Rule to estimate the distance traveled by the car.
(a) If f(x) = sin(sin x), use a graph to find an upper bound for |f(4)(x)|.(b) Use Simpson's Rule with n = 10 to approximateAnd use part (a) to estimate the error. (c) How large should n be to
Use the Comparison Theorem to determine whether the integral is convergent or divergent.a.b.
Find the area bounded by the curves y = cox x and y = cos2x between x = 0 and x = π.
The region under the curve y = cos2x, 0 ≤ x ≤ π/2, is rotated about the x-axis. Find the volume of the resulting solid.
If f' is continuous on [0, ˆž] and lim x†’ˆž f(x) = 0, show that
Use the substitution u = 1/x to show that
Three mathematics students have ordered a 14-inch pizza. Instead of slicing it in the traditional way, they decide to slice it by parallel cuts, as shown in the figure. Being mathematics majors, they
If 0 {ˆ«10 [bx + a(1 - x)]t dx}1/t.
Evaluate ∫∞ -1 (x4 / 1 + x6) 2 dx.
The circle with radius 1 shown in the figure touches the curve y = |2x| twice. Find the area of the region that lies between the two curves.
Evaluate ∫10 (2 √ 1 - x2 - 7 √ 1 - x3) dx.
An ellipse is cut out of a circle with radius a. The major axis of the ellipse coincides with a diameter of the circle and the minor axis has length 2b. Prove that the area of the remaining part of
A function f is defined byFind the minimum value of f.
Show thatStart by showing that if In denotes the integral, then
How do you evaluateIf m is odd? What if n is odd? What if m and n are both even?
If the expression √(a2 - x2) occurs in an integral, what substitution might you try? What if √(a2 + x2) occurs? What if √(a2 - x2) occurs?
Use the arc length formula 3 to find the length of the curve y = 2x - 5 -1 ≤ x ≤ 3. Check your answer by noting that the curve is a line segment and calculating its length by the distance
Find the length of the arc of the curve from point to point. y = 1/2 x2 P(-1, 1/2 ) Q(1, 1/2 )
Graph the curve and visually estimate its length. Then use your calculator to find the length correct to four decimal places. y = x2 + x3, 1 ≤ x ≤ 2
Use Simpson's Rule with to estimate the arc length of the curve. Compare your answer with the value of the integral produced by your calculator. a. y = x sin x, 0 ≤ x ≤ 2π b. y = ln ( 1 + x3). 0
(a) Graph the curve y = x 3ˆš(4-x), 0 ‰¤ x ‰¤ 4.(b) Compute the lengths of inscribed polygons with n = 1,2 and 4 sides. (Divide the interval into equal
Use either a computer algebra system or a table of integrals to find the exact length of the arc of the curve y = ln x that lies between the points (1,0) and (2, ln2).
a. y = sin x, 0 ≤ x ≤ π b. x = √y - y , 1 ≤ y ≤ 4
Sketch the curve with equation and use x 2/3 + y2/3 = 1 symmetry to find its length.
Find the arc length function for the curve with y = 2 x3/2 starting point Po (1,2) .
Find the arc length function for the curve y = sin-1 x + √(1 - x2) with starting point (0, 1).
A hawk flying at 15 m/s at an altitude of 180 m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation y = 180 - x2/45 Until it hits the ground, where
A manufacturer of corrugated metal roofing wants to produce panels that are 28 in. wide and 2 in. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the
Find the length of the curve1 ‰¤ x ‰¤ 4
Find the exact length of the curve. a. y = 1 + 6x3/2, 0 ≤ x ≤ 1 b. y = x3/3 + 1/4x, 1 ≤ x ≤ 2 c. x = 1/3 √y ( y - 3) , 1 ≤ y ≤ 9
(a) Set up an integral for the area of the surface obtained by rotating the curve about (i) the -axis and (ii) the -axis. (b) Use the numerical integration capability of your calculator to evaluate
The given curve is rotated about the -axis. Find the area of the resulting surface. a. y = 3√x, 1 ≤ y ≤ 2 b. x = √(a2 -y2), 0 ≤ y ≤ a/2
Use Simpson's Rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the -axis. Compare your answer with the value of the integral produced by your
Use either a CAS or a table of integrals to find the exact area of the surface obtained by rotating the given curve about the x-axis. y = 1/x , 1 ≤ x ≤ 2
Use a CAS to find the exact area of the surface obtained by rotating the curve about the -axis. If you're CAS has trouble evaluating the integral, express the surface area as an integral in the other
If the region R = = {(x, y) |x ‰¥ 1, 0 ‰¤ y ‰¤ 1/x} is rotated about the -axis, the volume of the resulting solid is finite. Show that the surface area is infinite.
(a) If a > 0, find the area of the surface generated by rotating the loop of the curve 3ay2 = x(a - x)2 about the -axis. (b) Find the surface area if the loop is rotated about the -axis.
(a) The ellipse x2/a2 + y2 /b2 = 1 a > b is rotated about the -axis to form a surface called an ellipsoid, or prolate spheroid. Find the surface area of this ellipsoid. (b) If the ellipse in part (a)
If the curve y = f(x), a ≤ x ≤ b, is rotated about the horizontal line y = c, where f(x) ≤ c, find a formula for the area of the resulting surface.
Find the area of the surface obtained by rotating the circle x2 + y2 = r2 about the line y = r.
Formula 4 is valid only when f(x) ‰¥ 0. Show that when f(x) is not necessarily positive, the formula for surface area becomes
Find the exact area of the surface obtained by rotating the curve about the -axis.a. y = x3, 0 ≤ x ≤ 2b. y = √(1 + 4x), 1 ≤ x ≤ 5
An aquarium 5 ft long, 2 ft wide, and 3 ft deep is full of water. Find (a) The hydrostatic pressure on the bottom of the aquarium, (b) The hydrostatic force on the bottom, (c) The hydrostatic force
A trough is filled with a liquid of density 840 kg.m3. The ends of the trough are equilateral triangles with sides 8 m long and vertex at the bottom. Find the hydrostatic force on one end of the
A cube with 20-cm-long sides is sitting on the bottom of an aquarium in which the water is one meter deep. Estimate the hydrostatic force on (a) The top of the cube (b) One of the sides of the cube.
A swimming pool is 20 ft wide and 40 ft long and its bottom is an inclined plane, the shallow end having a depth of 3 ft and the deep end, 9 ft. If the pool is full of water, estimate the hydro
A metal plate was found submerged vertically in sea water, which has density 64lb/ft3. Measurements of the width of the plate were taken at the indicated depths. Use Simpson's Rule to estimate the
Point-masses m1 are located on the axis as shown. Find the moment M of the system about the origin and the center of mass.
The masses m1 are located at the points P1. Find the moments Mx and My and the center of mass of the system. m1 = 4, m2 = 2, m3= 4 P1(2, -3), P2(-3, 1), P3(3,5)
Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordi nates of the centroid.a. y = 2x, y = 0, x = 1b. y = ex, y = 0 , x = 0, x = 1
Find the centroid of the region bounded by the given curves.a. y = x2, x = y2b. y = sinx, y = cosx, x = 0, x= π/4
A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then
Calculate the moments Mx and My and the center of mass of a lamina with the given density and shape.
Find the centroid of the region bounded by the curves y = x3 - x and y = x2 - 1. Sketch the region and plot the centroid to see if your answer is reasonable.
Prove that the centroid of any triangle is located at the point of intersection of the medians.
Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles (from Exercise 39) and using additivity of moments.
If is the -coordinate of the centroid of the region that lies under the graph of a continuous function f, where a ‰¤ x ‰¤ b, show that
Use the Theorem of Pappus to find the volume of the given solid. A cone with height and base radius
The marginal cost function C'(x) was defined to be the derivative of the cost function. (See Sections 3.7 and 4.7) The marginal cost of producing gallons of orange juice is C'(x) = 0.82 - 0.00003x +
If the amount of capital that a company has at time t is f (t) , then the derivative, f'(t) , is called the net investment flow. Suppose that the net investment flow is million √t dollars per year
Calculate.
The dye dilution method is used to measure cardiac output with 6 mg of dye. The dye concentrations, in mg/L, are modeled by c(t) = 20te-0.6t, 0 ≤ t ≤ 10, , where is measured in seconds. Find the
The graph of the concentration function c(t) is shown after a 7-mg injection of dye into a heart. Use Simpson's Rule to estimate the cardiac output.
A mining company estimates that the marginal cost of extracting tons of copper ore from a mine is 0.6 + 0.008x, measured in thousands of dollars per ton. Start-up costs are $100,000. What is the cost
A demand curve is given by ( = 450/(x+8). Find the consumer surplus when the selling price is $10
If a supply curve is modeled by the equation, p = 200 + 0.2x3/2 find the producer surplus when the selling price is $400.
A company modeled the demand curve for its product ( in dollars) by the equation P = (800000e-x/5000)/(x+20,000) Use a graph to estimate the sales level when the selling price is $16. Then find
Let f(x) be the probability density function for the lifetime of a manufacturer's highest quality car tire, where is measured in miles. Explain the meaning of each integral.a.b.
The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2.5 minutes. (a) Find the probability that a customer has to wait more than 4
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
The speeds of vehicles on a highway with speed limit 100km/h are normally distributed with mean 112km/h and standard deviation 8km/h. (a) What is the probability that a randomly chosen vehicle is
For any normal distribution, find the probability that the random variable lies within two standard deviations of the mean.
The hydrogen atom is composed of one proton in the nucleus and one electron, which moves about the nucleus. In the quantum theory of atomic structure, it is assumed that the electron does not move in
Let f(x) = 30x2(1-x)2 for 0 ≤ x ≤ 1 and f(x) = 0 for all other values of x. a. Verify that f is a probability density function. b. Find P(X ≤ 1/3)
Let f(x) = c/(1-x2) (a) For what value of is a probability density function? (b) For that value of c, find P(1- < X < 1).
A spinner from a board game randomly indicates a real number between 0 and 10. The spinner is fair in the sense that it indicates a number in a given interval with the same probability as it
Show that the median waiting time for a phone call to the company described in Example 4 is about 3.5 minutes.
Find the length of the curve. y = 1/6(x2 + 4)3/2 0 ≤ x ≤ 3
Find the centroid of the region bounded by the given curves. y = 1/2 x, y = √x
Find the centroid of the region shown below
Find the volume obtained when the circle of radius 1 with center (1, 0) is rotated about the -axis.
The demand function for a commodity is given by p = 2000 - 0.1x - 0.01x2 Find the consumer surplus when the sales level is 100.
(a) Explain why the functionis a probability density function. (b) Find P(X (c) Calculate the mean. Is the value what you would expect?
The length of time spent waiting in line at a certain bank is modeled by an exponential density function with mean 8 minutes. (a) What is the probability that a customer is served in the first 3
(a) Find the length of the curve y = x4/16 + 1/2x2, 1 ≤ x ≤ 2 b) Find the area of the surface obtained by rotating the curve in part (a) about the -axis.
Use Simpson's Rule with n=10 to estimate the length of the sine curve y = sinx, 0 ≤ x ≤ π
Find the length of the curve
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
Find the area of the region S = {(x, y) | x ≥ 0, y ≤ 1, x2 + y2 ≤ 4y}
In a famous 18th-century problem, known as Buffon's needle problem, a needle of length h is dropped onto a flat surface (for example, a table) on which parallel lines L units apart, L ‰¥ h
Find the centroid of the region enclosed by the ellipse x2 + (x + y + 1)2 = 1
If a sphere of radius is sliced by a plane whose distance from the center of the sphere is d, then the sphere is divided into two pieces called segments of one base. The corresponding surfaces are
Suppose that the density of seawater, ( = ((z), varies with the depth below the surface. (a) Show that the hydrostatic pressure is governed by the differential equation dp/dz = ((z)g Where is the
Let P be a pyramid with a square base of side 2b and suppose that is a sphere with its center on the base P of S and is tangent to all eight edges of P. Find the height of P. Then find the volume of

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