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mathematics
calculus

Questions and Answers of Calculus

Find the arc length function for the curve y = 2x3/2 with starting point P0 (1, 2).
(a) Graph the curve y = 1/3x3 + 1/(4x), x > 0.(b) Find the arc length function for this curve with starting point P0 (1, 1/12).(c) Graph the arc length function.
A hawk flying at 15m/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 steady wind blows a kite due west. The kite’s height above ground from horizontal position x = 0 to x = 80ft is given by y = 150 – 1/40 (x – 50)2 Find the distance traveled by the kite.
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
(a) The figure shows a telephone wire hanging between two poles at x = €“ b and x = b. It takes the shape of a catenary's with equation y = c + a cosh (x/a). Find the length of the wire.(b)
Find the length of the curve y = ∫x √t3 – 1 dt, 1 < x < 4.
The curves with equations xn + yn = 1, n = 4, 6, 8, . . . , are called fat circles. Graph the curves with n = 2, 4, 6, 8, and 10 to see why. Set up an integral for the length L2k of the fat circle
Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve about the given axis.
Find the area of the surface obtained by rotating the curve about the -axis.
The given curve is rotated about the -axis. Find the area of the resulting surface.
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.
Use a CAS to find the exact area of the surface obtained by rotating the curve about the-axis, if your 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
If the infinite curve y = e–x, x > 0, is rotated about the -axis, CAS find the area of the resulting surface.
(a) If a > 0, find the area of the surface generated by rotating the loop of the curve 2ay2 = x(a – x)2 about the y-axis.(b) Find the surface area if the loop is rotated about the y-axis.
A group of engineers is building a parabolic satellite dish whose shape will be formed by rotating the curve y = ax2 about the -axis. If the dish is to have a 10-ft diameter and a maximum depth of 2
The ellipse is rotated about the -axis to form a surface called an ellipsoid.Find the surface area of this ellipsoid.
Find the surface area of the torus in Exercise 61 in Section 6.2.
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.
Use the result of Exercise 31 to set up an integral to find the area of the surface generated by rotating the curve y = √x, 0 < x < 4, about the line y = 4. Then use a CAS to evaluate the
Find the area of the surface obtained by rotating the circle x2 + y2 = r2 about the line y = r.
Show that the surface area of a zone of a sphere that lies between two parallel planes is S = πdh, where is the diameter of the sphere and is the distance between the planes. (Notice that
Formula 4 is valid only when f(x) > 0. Show that when f(x) is not necessarily positive, the formula for surface area becomes
Let L be the length of the curve y = f(x), a < x < b, where f is positive and has a continuous derivative. Let Sf be the surface area generated by rotating the curve about the x-axis. If c is a
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, and (c) The hydrostatic
A swimming pool 5 m wide, 10 m long, and 3 m deep is filled with seawater of density 1030 kg/m3 to a depth of 2.5 m. Find(a) The hydrostatic pressure at the bottom of the pool, (b) The hydrostatic
A vertical plate is submerged in water and has the indicated shape. Explain how to approximate the hydrostatic force against the end of the tank by a Riemann sum. Then express the force as an
A large tank is designed with ends in the shape of the region between the curves y = x2/2 and y = 12, measured in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of
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 vertical dam has a semicircular gate as shown in the figure. Find the hydrostatic force against the gate.
A cube with 20-cm-long sides is sitting on the bottom of an aquarium in which the water is one meter deep. Find the hydrostatic force on (a) The top of the cube and (b) One of the sides of the cube.
A dam is inclined at an angle of 30 from the vertical and has the shape of an isosceles trapezoid 100 ft wide at the top and 50 ft wide at the bottom and with a slant height of 70 ft, find the
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, find the hydrostatic
Suppose that a plate is immersed vertically in a fluid with density and the width of the plate is w(x) at a depth of meters beneath the surface of the fluid. If the top of the plate is at depth a and
A vertical, irregularly shaped plate is submerged in water. The table shows measurements of its width, taken at the indicated depths. Use Simpson€™s rule to estimate the force of the water
(a) Use the formula of Exercise 16 to show that F = (pgx) A where is the -coordinate of the centroid of the plate and A is its area. This equation shows that the hydrostatic force against a vertical
Point-masses mi are located on the -axis as shown. Find the moment M of the system about the origin and the center of mass x.
The masses mi are located at the points P. Find the moments Mx and My and the center of mass of the system.
Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.
Find the centroid of the region bounded by the given curves.
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 = 2x and y = x2, 0 < x < 2, to three decimal places. Sketch the region and plot the centroid to see if your answer is reasonable.
Use a graph to find approximate -coordinates of the points of intersection of the curves y = x + in x and y = x3 – x. Then find (approximately) the centroid of the region bounded by these curves.
Prove that the centroid of any triangle is located at the point of intersection of the medians. [Hints: Place the axes so that the vertices are (a, 0), (0, b), and (c, 0). Recall that a median is a
Find the centroid of the region shown, not by integration, but by locating the centroid's of the rectangles and triangles (from Exercise 37) and using additively of moments.
Use the Theorem of Pappus to find the volume of the given solid.
Let R be the region that lies between the curves y = xm and y = xn, 0 < x < 1, where m and are integers with 0 < n < m.(a) Sketch the region R.(b) Find the coordinates of the centroid of R.(c) Try to
The marginal cost function C’(x) was defined to be the derivative of the cost function. (See Sections 3.3 and 4.8) If the marginal cost of manufacturing meters of a fabric is C’(x) = 5 – 0.008x
The marginal revenue from the sale of units of a product is 12 – 0.0004x. If the revenue from the sale of the first 1000 units is $12,400, find the revenue from the sale of the first 5000 units.
The marginal cost of producing units of a certain product is 74 + 1.1x – 0.002x2 + 0.00004x3 (in dollars per unit). Find the increase in cost if the production level is raised from 1200 units to
The demand function for a certain commodity is p = 5 – x/10. Find the consumer surplus when the sales level is 30. Illustrate by drawing the demand curve and identifying the consumer surplus as an
A demand curve is given by p = 450/(x + 8). Find the consumer surplus when the selling price is $10.
The supply function ps(x) for a commodity give the relation between the selling price and the number of units that manufacturers will produce at that price. For a higher price, manufacturers will
If a supply curve is modeled by the equation p 200 + 0.2x3/2, find the producer surplus when the selling price is $400.
For a given commodity and pure competition, the number of units produced and the price per unit are determined as the coordinates of the point of intersection of the supply and demand curves. Given
A company modeled the demand curve for its product (in dollars) byUse a graph to estimate the sales level when the selling price is $16. Then find (approximately) the consumer surplus for this sales
A movie theater has been charging $7.50 per person and selling about 400 tickets on a typical weeknight. After surveying their customers, the theater estimates that for every 50 cents that they lower
If the amount of capital that a company has at time is f(t), then the derivative, f’(t), is called the net investment flow. Suppose that the net investment flow is √t million dollars per year
A hot, wet summer is causing a mosquito population explosion in a lake resort area. The number of mosquitoes is increasing at an estimated rate of 2200 + 10e0.8t per week (where is measured in
Use Poiseuille’s Law to calculate the rate of flow in a small human artery where we can take η = 0.027, R = 0.008 cm, l = 2 cm, and P = 4000 dynes/cm2.
High blood pressure results from constriction of the arteries. To maintain a normal flow rate (flux), the heart has to pump harder, thus increasing the blood pressure. Use Poiseuille€™s Law
The dye dilution method is used to measure cardiac output with 8 mg of dye. The dye concentrations, in mg/L, are modeled by c(t) = ¼ t(12 – t), 0 < t < 12, where is measured in seconds. Find the
After an 8-mg injection of dye, the readings of dye concentration at two-second intervals are as shown in the table. Use Simpson€™s Rule to estimate the cardiac output.
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.
Let f(x) be the probability density function for the time it takes you to drive to school in the morning, where is measured in minutes. Express the following probabilities as integrals.(a) The
Let f(x) = 3/64 x√16 – x2 for 0 < x < 4 and f(x) = 0 for all other values of x. (a) Verify that f is a probability density function. (b) Find P(X < 2).
Let f(x) = kx2 (1 – x) if 0 < x < 1 and f(x) = 0 if x < 0 or x > 1.(a) For what value of k is f a probability density function?(b) For that value of k, find P(X > ½).(c) Find the mean.
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
(a) Explain why the function whose graph is shown is a probability density function.(b) Use the graph to find the following probabilities:(i) P (X (c) Calculate the mean.
Show that the median waiting time for a phone call to the company described in Example 4 is about 3.5 minutes.
(a) A type of lightbulb is labeled as having an average lifetime of 1000 hours. It’s reasonable to model the probability of failure of these bulbs by an exponential density function with mean
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 for more than 4
According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches.(a) What is the probability
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
Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g.(a) If the target weight is 500 g, what is
For any normal distribution, find the probability that the random variable lies within two standard deviations of the mean.
The standard deviation for a random variable with probability density function f and mean μ is defined byFind the standard deviation for an exponential density function with mean μ.
(a) Write an expression for the surface area of the surface obtained by rotating the curve y = f(x), a < x < b, about the x-axis.(b) What if is given as a function of y?(c) What if the curve is
Describe how we can find the hydrostatic force against a vertical wall submersed in a fluid.
What does the Theorem of Pappus say?
What is a probability density function? What properties does such a function have?
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 ∫100 f(x) dx? (b) Write an
Find the length of the curve.
(a) Find the length of the curve(b) Find the area of the surface obtained by rotating the curve in part (a) about the -axis.
(a) The curve y = x2, 0 < x < 1, is rotated about the -axis. Find the area of the resulting surface.(b) Find the area of the surface obtained by rotating the curve in part (a) about the x-axis.
Use Simpson’s Rule with n = 6 to estimate the length of the curve y = e–x2, 0 < x < 3.
Use Simpson’s Rule with n = 6 to estimate the area of the surface obtained by rotating the curve in Exercise 5 about the x-axis.
Find the length of the curve
Find the area of the surface obtained by rotating the curve in Exercise 7 about the y-axis.
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, with the water extending to
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.
Find the centroid of the region bounded by the given curves.
Find the centroid of the region shown.
Find the volume obtained when the circle of radius 1 with center (1, 0) is rotated about the y-axis.
Use the Theorem of Pappus and the fact that the volume of a sphere of radius r is 4/3 πr3 to find the centroid of the semicircular region bounded by the curve y = √r2 – x2 and the x-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.
After a 6-mg injection of dye into a heart, the readings of dye concentration at two-second intervals are as shown in the table. Use Simpson€™s Rule to estimate the cardiac output.
(a) Explain why the function is a probability density function.(b) Find P(X (c) Calculate the mean. Is the value what you would expect?

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