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mathematics
calculus

Questions and Answers of Calculus

(a) Show that the volume of a segment of height h of a sphere of radius r isV = 1/3 Ï€h2 (3r - h)
Water in an open bowl evaporates at a rate proportional to the area of the surface of the water. (This means that the rate of decrease of the volume is proportional to the area of the surface.) Show
The figure shows a curve C with the property that, for every point P on the middle curve y = 2x2, the areas A and B re equal. Find an equation for C.
(a) What is the volume of a cylindrical shell? (b) Explain how to use cylindrical shells to find the volume of a solid of revolution. (c) Why might you want to use the shell method instead of slicing?
(a) What is the average value of a function f on an interval [a, b]? (b) What does the Mean Value Theorem for Integrals say? What is its geometric interpretation?
Evaluate the integral using integration by parts with the indicated choices of u and dv. ∫ x2 ln x dx u = ln x dv = x2 dx
Evaluate the integral. a. ∫ x cos 5x dx b. ∫ te-3t dt c. ∫ (x2 +2x) cos x dx
First make a substitution and then use integration by parts to evaluate the integral.a. ˆ« cos ˆšx dxc. ˆ« x ln (1 + x) dx
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its anti derivative (take C = 0). a. ∫xe-2x dx b. ∫ x3 √(1+x2) dx
(a) Use the reduction formula in Example 6 to show that ∫ sin2x dx = x/2 - sin 2x/4 + C (b) Use part (a) and the reduction formula to evaluate ∫ sin4x dx
(a) Use the reduction formula in Example 6 to show thatWhere n ‰¥ 2 is an integer (b) Use part (a) to evaluate
Use integration by parts to prove the reduction formula. a. ∫ (ln x)n dx = x(ln x)n - n ∫ (ln x)n-1 dx b. ∫ tannx dx = = (tann-1 x)/(n - 1) - ∫ tann-2 x dx (n ≠ 1)
Use Exercise 51 to find ∫ (lnx)3dx. In Exercise 51 ∫ (ln x)n dx = x(ln x)n - n ∫ (ln x)n-1 dx
Find the area of the region bounded by the given curves. y = x2 ln x ....................... y = 4 ln x
Use a graph to find approximate -coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. y = arcsin (1/2x)
Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. a. y = cos (πx/2), y = 0, 0 ≤ x ≤ 1 about the
Calculate the average value of f(x) = x sec2x on the interval [0, π/4]
A particle that moves along a straight line has velocity v(t) = t2e-t meters per second after seconds. How far will it travel during the first seconds?
Suppose that f(1) = 2, f(4) = 7, f'(1) = 5, f'(4) = 3, and f'' is continuous. Find the value of
We arrived at Formula 6.3.2,by using cylindrical shells, but now we can use integration by parts to prove it using the slicing method of Section 6.2, at least for the case where f is one-to-one and
Evaluate the integral.a. ˆ« sin2x cos3 x dxb.c.
Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its anti derivative (taking C = 0). a. ∫ x sin2 (x2) dx b. ∫ sin3x sin
Find the average value of the function f(x) = sin2x cos3x on the interval [- π, π]
Find the area of the region bounded by the given curves. y = sin2x, y = cos2x, -π/4 ≤ x ≤ π/4
Use a graph of the integrand to guess the value of the integral. Then use the methods of this section to prove that your guess is correct.
Find the volume obtained by rotating the region bounded by the given curves about the specified axis. a. y = sin x, y = 0 , π/2 ≤ x ≤ π about the x-axis. b. y = sinx, y = cos x, 0 ≤ x ≤ π
A particle moves on a straight line with velocity function v(t) = sin ωt cos2 ωt. . Find its position function s= f(t) if f(0) = 0
Prove the formula, where and are positive integers.a.b.
Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. a. ∫ dx/(x2 √4 - x2) ................... x = 2sinθ b. ∫ √(x2 - 4)/x
(a) Use trigonometric substitution to show that ∫ dx/√(x2+a2) = ln(x+√(x2+a2) + C (b) Use the hyperbolic substitution to show that ∫ dx/√(x2 + a2) = sinh-1(x/a) + C These formulas are
Find the average value of f(x) = √(x2 -1)/x, 1 ≤ x ≤ 7.
Prove the formula A = 1/2 r2Î¸ for the area of a sector of a circle with radius r and central angle Î¸. [Assume 0
Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y = 9/(x2 + 9). y = 0, x = 0, and x = 3
(a) Use trigonometric substitution to verify that(b) Use the figure to give trigonometric interpretations of both terms on the right side of the equation in part (a).
A torus is generated by rotating the circle x2 + (y - R)2 = r2 about the x-axis. Find the volume enclosed by the torus.
Find the area of the crescent-shaped region bounded by arcs of circles with radii r and R. (See the figure.)
Evaluate the integral.a.b.
Write out the form of the partial fraction decomposition of the function. Do not determine the numerical values of the coefficients. 1. (a). (1 + 6x)/((4x - 3)(2x + 5)) (b). 10/(5x2 - 2x3) 2. (a).
Make a substitution to express the integrand as a rational function and then evaluate the integral. a. ∫ √(x+1)/x dx b. ∫ dx/(x2+x√x
Use integration by parts, together with the techniques of this section, to evaluate the integral. ∫ ln(x2 - x + 2) dx
Use a graph of f(x) = 1/(x2 -2x -3) to decide whetheris positive or negative. Use the graph to give a rough estimate of the value of the integral and then use partial fractions to find the exact
Evaluate the integral by completing the square and using Formula 6. ∫ dx/(x2 - 2x)
The German mathematician Karl Weierstrass (1815-1897) noticed that the substitution t = tan(x/2) will convert any rational function of sin x and cos x into an ordinary rational function of t. (a) If
Use the substitution in Exercise 59 to transform the integrand into a rational function of and then evaluate the integrala. ˆ« 1/(3sinx - 4cosx) dxb.
Find the area of the region under the given curve from 1 to 2. y = (x2 + 1)/(3x - x2)
One method of slowing the growth of an insect population without using pesticides is to introduce into the population a number of sterile males that mate with fertile females but produce no
(a) Use a computer algebra system to find the partial fraction decomposition of the function(b) Use part (a) to find ˆ« f(x) dx (by hand) and compare with the result of using the CAS to
Evaluate the integral. a. ∫ x4/(x - 1) dx b. ∫ (5x + 1)/((2x + 1)(x - 1)) dx
Suppose that F, G, and Q are polynomials and F(x)/Q(x) = G(x)/Q(x) for all except when Q(x) = 0. Prove that F(x) = G(x) for all x.
If a ≠ 0 and is a positive integer, find the partial fraction decomposition of f(x) = 1/xn(x-a) First find the coefficient of 1/(x-a). Then subtract the resulting term and simplify what is left
Evaluate the integral. a. ∫ cos x (1 + sin2x) dx b. ∫ (sinx + secx)/tanx dx c. ∫ t/(t4+2)
The functionsAndDon't have elementary antiderivatives, butDoes. Evaluate
Use the indicated entry in the Table of Integrals on the Reference Pages to evaluate the integral.a.Entry 80 b. Entry 39
The region under the curve y = sin2x from 0 to π is rotated about the x-axis. Find the volume of the resulting solid.
Verify Formula 53 in the Table of Integrals (a) by differentiation and (b) by using the substitution t = a + bu.
Use a computer algebra system to evaluate the integral. Compare the answer with the result of using tables. If the answers are not the same, show that they are equivalent. a. ∫ sec4xdx b. ∫ x2
(a) Use the table of integrals to evaluate F(x) = ∫ f(x) dx, where f(x) = 1/(x√(1 - x2)) What is the domain of f and F? (b) Use a CAS to evaluate F(x). What is the domain of the function that the
Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.a.b. c. ˆ« dx/(x2 ˆš(4x2 + 9))
LetWhere f is the function whose graph is shown (a) Use the graph to find L2, R2 and M2. (b) Are these underestimates or overestimates of I? (c) Use the graph to find T2. How does it compare with
(a) Find the approximations TS and MS for the integral(b) Estimate the errors in the approximations of part (a). (c) How large do we have to choose n so that the approximations Tn and Mn to the
(a) Find the approximations T10, M10 and S10 forAnd the corresponding errors ET, EM, and ES (b) Compare the actual errors in part (a) with the error estimates given by 3 and 4. (c) How large do we
The trouble with the error estimates is that it is often very difficult to compute four derivatives and obtain a good upper bound K for | f(4) | by hand. But computer algebra systems have no problem
Find the approximations Ln, Rn, Tn, and Mn for n = 5, 10, and 20. Then compute the corresponding errors EL, ER, ET, and EM. (Round your answers to six decimal places you may wish to use the sum
Find the approximations Tn, Mn, and Sn for n = 6 and 12. Then compute the corresponding errors ET, EM and ES. (Round your answers to six decimal places you may wish to use the sum command on a
Estimate the area under the graph in the figure by using(a) The Trapezoidal Rule,(b) The Midpoint Rule, and(c) Simpson's Rule, each with
EstimateUsing (a) The Trapezoidal Rule and (b) The Midpoint Rule, each with n = 4. From a graph of the integrand, decide whether your answers are underestimates or overestimates. What can you
a) Use the Midpoint Rule and the given data to estimate the value of the integralb) If it is known that -2 ‰¤ f''(x) ‰¤ 3 for all x, estimate the error involved in the
A graph of the temperature in New York City on September 19, 2009 is shown. Use Simpson's Rule with n = 12 to estimate the average temperature on that day.
The graph of the acceleration a(t) of a car measured in ft/s is shown. Use Simpson's Rule to estimate the increase in the velocity of the car during the 6-second time interval.
The table (supplied by San Diego Gas and Electric) gives the power consumption P in megawatts in San Diego County from midnight to 6:00 AM on a day in December. Use Simpson's Rule to estimate the
Use Simpson's Rule with n = 8 to estimate the volume of the solid obtained by rotating the region shown in the figure about(a) The x-axis(b) The y-axis.
The region bounded by the curves y = e-1/x, y = 0, x = 1, and x = 5 is rotated about the x-axis. Use Simpson's Rule with n = 8 to estimate the volume of the resulting solid.
The intensity of light with wavelength traveling through a diffraction grating with N slits at an angle Î¸ is given by I(Î¸) = N2sin2k/k2, where and K =
Sketch the graph of a continuous function [0, 2] on for which the Trapezoidal Rule with n = 2 is more accurate than the Midpoint Rule.
If f is a positive function and f''(x)
Show that 1/2 (Tn + Mn) T2n.
Use(a) The Midpoint Rule(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
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.n = 10 2. n =
Explain why each of the following integrals is improper.a.b. c. d.
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.
Sketch the region and find its area (if the area is finite). a. S = {(x, y) x ≥ 1, 0 ≤ y ≤ e-x} b. S = {(x, y) x ≥ 1, 0 ≤ y ≤ 1/(x3 + x)}
(a) If g(x) = (sin2x)/x2, use your calculator or computer to make a table of approximate values offor t = 2, 5, 10, 100, 1000, and 10,000. Does it appear that is convergent? (b) Use the Comparison
Use the Comparison Theorem to determine whether the integral is convergent or divergent.a.b.
Determine whether each integral is convergent or divergent. Evaluate those that are convergent.a.b. c.
The integralis improper for two reasons: The interval [0, ˆž] is infinite and the integrand has an infinite discontinuity at 0. Evaluate it by expressing it as a sum of improper integrals
Find the values of p for which the integral converges and evaluate the integral for those values of p.
a. Show thatis divergent. b Show that This shows that we can't define
We know from Example 1 that the region R = {(x, y) | x ≥ 1, 0 ≤ y ≤ 1/x} has infinite area. Show that by R rotating about the -axis we obtain a solid with finite volume.
Find the escape velocity v0 that is needed to propel a rocket of mass m out of the gravitational field of a planet with mass M and radius R. Use Newton's Law of Gravitation and the fact that the
A manufacturer of light bulbs 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
Determine how large the number has to be so that
If f(t) is continuous for t ‰¥ 0, the Laplace transform of f is the function F defined byand the domain of F is the set consisting of all numbers s for which the integral converges. Find the
Suppose that 0 ≤ f(t) ≤ Meat and 0 ≤ f'(t) ≤ Keat for t ≥ 0, where f' is continuous. If the Laplace transform of f(t) is F(s) and the Laplace transform of f'(t) is G(s), show that G(s) =
Show that
Find the value of the constant C for which the integralconverges. Evaluate the integral for this value of C.
Suppose is continuous on [0, ˆž] and limx†’ˆž f(x) = 1. Is it possible that is convergent?
Evaluate the integral.a.b. c.
Evaluate the integral or show that it is divergent.a.b.
Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0).
Graph the function f(x) = cos2xsin3x and use the graph to guess the value of the integral.Then evaluate the integral to confirm your guess.

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