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mathematics
calculus 4th

Questions and Answers of Calculus 4th

Define the quartiles of a probability function to be those values a1, a2, and a3 such that P(−∞ < x ≤ a1) = P(a1 ≤ x ≤ a2) = P(a2 ≤ x ≤ a3) = P(a3 ≤ x < ∞) = 14. Find the quartile
Generalize the result of Exercise 25 to show that the circumference of the circle of radius r is 2πr.Data From Exercise 25Show that the circumference of the unit circle is equal toEvaluate, thus
Calculate μ and σ, where σ is the standard deviation, defined byThe smaller the value of σ, the more tightly clustered are the values of the random variable X about the mean μ. (The limits of
The end of the trough in Figure 19 is an equilateral triangle of side 3. Assume that the trough is filled with water to height H. Calculate the fluid force on each side of the trough as a function of
Calculate μ and σ, where σ is the standard deviation, defined by o² = (x-μ)²p(x) dx -00
Calculate the arc length of y = x2 over [0, a]. Use trigonometric substitution. Evaluate for a = 1.
Calculate μ and σ, where σ is the standard deviation, defined by o² = (x-μ)² p(x) dx -00
A rectangular plate of side ℓ is submerged vertically in a fluid of density w, with its top edge at depth h. Show that if the depth is increased by an amount Δh, then the force on a side of the
Express the arc length of g(x) = √x over [0, 1] as a definite integral. Then use the substitution u = √x to show that this arc length is equal to the arc length of y = x2 over [0, 1] (but do not
Calculate μ and σ, where σ is the standard deviation, defined by o² = *(x-μ)²p(x) dx -00
Prove that the force on the side of a rectangular plate of area A submerged vertically in a fluid is equal to p0A, where p0 is the fluid pressure at the center point of the rectangle.
Find the arc length of y = ex over [0, a]. Try the substitution u = √1 + e2x followed by partial fractions.
Show that the arc length of y = ln(ƒ(x)) for a ≤ x ≤ b is nb a f(x)² + f'(x)² -dx f(x)
Use Eq. (3) to compute the arc length of y = ln(sin x) for π/4 ≤ x ≤ π/2 a b √√f(x)² + f'(x)² f(x) - dx
The time to decay of an atom in a radioactive substance is a random variable X. The law of radioactive decay states that if N atoms are present at time t = 0, then N ƒ(t) atoms will be present at
Use Eq. (3) to compute the arc length of y = ln (ex + 1)/(ex − 1) over [1, 3]. b √√f(x)² + f'(x)² f(x) - dx
The half-life of radon-222 is 3.825 days. Use Exercise 31 to compute:(a) The average time to decay of a radon-222 atom.(b) The probability that a given atom will decay in the next 24 hours.Data From
Use the Comparison Theorem (Section 5.2) to prove that the arc length of y = x4/3 over [1, 2] is not less than 5/3. THEOREM 5 Comparison Theorem If f and g are integrable and g(x) ≤ f(x) for x in
Show that if 0 ≤ ƒ(x) ≤ 1 for all x, then the arc length of y = ƒ(x) over [a, b] is at most √2(b − a). Show that for ƒ(x) = x, the arc length equals √2(b − a).
Approximate the arc length of one-quarter of the unit circle (which we know is π/2) by computing the length of the polygonal approximation with N = 4 segments (Figure 17). y 1 0.25 0.5 0.75 1 -X
A merchant intends to produce specialty carpets in the shape of the region in Figure 18, bounded by the axes and graph of y = 1 − xn (units in yards). Assume that material costs $50/yd2 and that it
Compute the surface area of revolution about the x-axis over the interval.y = x, [0, 4]
Compute the surface area of revolution about the x-axis over the interval.y = 4x + 3, [0, 1]
Compute the surface area of revolution about the x-axis over the interval.y = x3, [0, 2]
Compute the surface area of revolution about the x-axis over the interval.y = x3, [0, 10]
Compute the surface area of revolution about the x-axis over the interval.y = x2, [0, 2]
Compute the surface area of revolution about the x-axis over the interval.y = x2, [0, 10]
Compute the surface area of revolution about the x-axis over the interval.y = (4 − x2/3)3/2, [0, 8]
Compute the surface area of revolution about the x-axis over the interval. y=x²-lnx, [1, e]
Compute the surface area of revolution about the x-axis over the interval.y = e−x, [0, 1]
Compute the surface area of revolution about the x-axis over the interval.y = sin x, [0, π]
Use a computer algebra system to find the approximate surface area of the solid generated by rotating the curve about the x-axis.y = x−1, [1, 3]
Use a computer algebra system to find the approximate surface area of the solid generated by rotating the curve about the x-axis.y = x4, [0, 1]
Use a computer algebra system to find the approximate surface area of the solid generated by rotating the curve about the x-axis.y = e−x2/2, [0, 2]
Use a computer algebra system to find the approximate surface area of the solid generated by rotating the curve about the x-axis.y = tan x, [0, π/4]
Show that a spherical cap of height h and radius R (Figure 19) has surface area 2πRh. h R
Find the area of the surface obtained by rotating y = cosh x over [− ln 2, ln 2] around the x-axis.
Find the surface area of the torus obtained by rotating the circle x2 + (y − b)2 = r2 about the x-axis (Figure 20). (0, b+r) (0, b) -X
The graph of y = ƒ(x), for a ≤ x ≤ b, is rotated about the y-axis. In this situation, the surface area of the resulting surface isDetermine the surface area for each surface of revolution. If
The graph of y = ƒ(x), for a ≤ x ≤ b, is rotated about the y-axis. In this situation, the surface area of the resulting surface isDetermine the surface area for each surface of revolution. If
The graph of y = ƒ(x), for a ≤ x ≤ b, is rotated about the y-axis. In this situation, the surface area of the resulting surface isDetermine the surface area for each surface of revolution. If
The graph of y = ƒ(x), for a ≤ x ≤ b, is rotated about the y-axis. In this situation, the surface area of the resulting surface isDetermine the surface area for each surface of revolution. If
The graph of y = ƒ(x), for a ≤ x ≤ b, is rotated about the y-axis. In this situation, the surface area of the resulting surface isDetermine the surface area for each surface of revolution. If
Let L be the arc length of the upper half of the ellipse with equation(Figure 21) and let η = 1 − (b2/a2). Use substitution to show thatUse a computer algebra system to approximate L for a = 2, b
Find the surface area of the ellipsoid obtained by rotating the ellipse (x/a)2 + (y/b)2 = 1 about the x-axis.
Show that if the arc length of y = ƒ(x) over [0, a] is proportional to a, then y = ƒ(x) must be a linear function.
Let ƒ be an increasing function on [a, b] and let g be its inverse. Argue on the basis of arc length that the following equality holds:Then use the substitution u = ƒ(x) to prove Eq. (4). f(b) S*
Suppose that the observer in Exercise 62 moves off to infinity—that is, d → ∞. What do you expect the limiting value of the observed area to be? Check your guess by using the formula for the
Prove that the portion of a sphere of radius R seen by an observer located at a distance d above the North Pole has area A = 2πdR2/(d + R). According to Exercise 52, the cap has surface area 2πRh.
Let M be the total mass of a metal rod in the shape of the curve y = ƒ(x) over [a, b] whose mass density ρ(x) varies as a function of x. Use Riemann sums to justify the formula M = √² p(x) √ 1
Which of the following integrals is improper? Explain your answer, but do not evaluate the integral. (a) (d) (j) S² So' S dx x1/3 ex dx sin x dx Linx In x dx (b) (e) (h) S C dx +0.2 sec x
Compute the work (in joules) required to stretch or compress a spring as indicated, assuming a spring constant of k = 800 N/m.Compressing 4 cm more when it is already compressed 5 cm
Match the integrals (a)–(e) with their antiderivatives (i)–(v) on the basis of the general form (do not evaluate the integrals). xdx (a) √x²-4 ·S (c) fs sin³ x cos² x dx 16 dx (e)
Calculate MN and TN for the value of N indicated and compare with the actual value of the integral. S [₁²₁x² dx, N = 4
What are T1 and T2 for a function on [0, 2] such that ƒ(0) = 3, ƒ(1) = 4, and ƒ(2) = 3?
Let ƒ(x) = x−4/3. • St (a) Evaluate (b) Evaluate f(x) dx. f(x) dx by computing the limit S lim R→∞0 f(x) dx
For which graph in Figure 15 will TN overestimate the integral? What about MN? y = f(x) x y y = g(x) -X
Calculate MN and TN for the value of N indicated and compare with the actual value of the integral. S √xdx, N = 4
Prove that ∫∞1 x−2/3 dx diverges by showing that R S² lim R→∞0 x-2/3 dx = ∞
Evaluate x dx/x + 2 in two ways: using substitution and using the Method of Partial Fractions.
Calculate MN and TN for the value of N indicated and compare with the actual value of the integral. [²x³dx, N = 6
How large is the error when the Trapezoidal Rule is applied to a linear function? Explain graphically
Evaluate using the suggested method. - [ cos³ e sin³ e de [Write cos³ e as cos 0(1 – sin² e).]
What is the maximum possible error if T4 is used to approximatewhere |ƒ"(x)| ≤ 2 for all x? f(x) dx
Calculate MN and TN for the value of N indicated and compare with the actual value of the integral. S² ex dx, N = 6
Evaluate using the suggested method. -12x dx e-12x dx (Integration by Parts) Sxe
Determine whether the improper integral converges and, if so, evaluate it. 00 J1 xp x19/20
Calculate MN and TN for the value of N indicated. S dx X N = 6
What are the two graphical interpretations of the Midpoint Rule?
Evaluate using the suggested method. S sec sec³ tan de (trigonometric identity, reduction formula)
Determine whether the improper integral converges and, if so, evaluate it. 00 S: dx x20/19
Calculate MN and TN for the value of N indicated. √₁² √x+ +1dx, N=5
Evaluate using the suggested method. S - ² 4x + 4 (x - 5)(x+3) dx (partial fractions)
Determine whether the improper integral converges and, if so, evaluate it. 1P 110000 00- 8
Calculate MN and TN for the value of N indicated. r/2 Vsin x dx, N = 6
Evaluate using the suggested method. 1 S - dx (trigonometric substitution) x(x² - 1)3/2
Determine whether the improper integral converges and, if so, evaluate it. 00 dt Sof t 20
Calculate MN and TN for the value of N indicated. 0 /4 sec xdx, N = 6
Evaluate using the suggested method. [(1 + x²)-3/2 dx (trigonometric substitution)
Determine whether the improper integral converges and, if so, evaluate it. S 0 dx x20/19
Calculate MN and TN for the value of N indicated. J1 Inxdx, N = 5
Evaluate using the suggested method. dx x3/2 + x1/2 (substitution)
Determine whether the improper integral converges and, if so, evaluate it. S dx x19/20
Calculate MN and TN for the value of N indicated. dx Si J₂ ln x N = 5
Evaluate using the suggested method. dx x + x¹ S= (rewrite integrand)
Determine whether the improper integral converges and, if so, evaluate it. 10 dx √4-x
Calculate MN and TN for the value of N indicated. Se-x² dx e-² dx, N = 5
Evaluate using the suggested method. S₁ x2 tan¹ xdx (Integration by Parts)
Determine whether the improper integral converges and, if so, evaluate it. dx 5 (x - 5)³/2 5.
Calculate MN and TN for the value of N indicated. -2 dx, N = 6
Evaluate using the suggested method. dx x² + 4x5 Sz (complete the square, substitution, partial fractions)
Determine whether the improper integral converges and, if so, evaluate it. 00 5₁²x 2 x³ dx
Calculate SN given by Simpson’s Rule for the value of N indicated and compare with the actual value of the integral. S Ảdx, N=4
Evaluate using the appropriate method or combination of methods. S x²e4x dx
Determine whether the improper integral converges and, if so, evaluate it. 00€ dx 0 (x + 1)³ S.
Calculate SN given by Simpson’s Rule for the value of N indicated and compare with the actual value of the integral. S √xdx, N = 4
Evaluate using the appropriate method or combination of methods. S x² V9 - x² dx
Determine whether the improper integral converges and, if so, evaluate it. 00 S J-3 dx (x+4)³/2
Calculate SN given by Simpson’s Rule for the value of N indicated and compare with the actual value of the integral. Se ex dx, N = 6
Evaluate using the appropriate method or combination of methods. I cos cos 60 sin³ 60 de

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