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mathematics
calculus 10th edition
Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions
In Exercises, find the indefinite integral. J sin 4x dx
In Exercises use a computer algebra system and the error formulas to find n such that the error in the approximation of the definite integral is less than or equal to 0.00001 using (a) The Trapezoidal Rule (b) Simpson's Rule. So sin x² dx
Approximate the area of the shaded region using (a) The Trapezoidal Rule with n = 4. (b) Simpson's Rule with n = 4. 10 8 00 4 2 y 1 2 3 4 5 X
In Exercises, find the indefinite integral. I cos cos 8x dx
Approximate the area of the shaded region using(a) The Trapezoidal Rule with n = 8.(b) Simpson's Rule with n = 8. 10 8 6 4 2 y 7 XH 2 4 6 8 10 X
Use Simpson's Rule with n = 14 to approximate the area of the region bounded by the graphs of y = √x cos x, y = 0, x = 0, and x = π/2.
In Exercises, find the indefinite integral. [esc² (1) dx
In Exercises, find the indefinite integral. 0² cos - de 7 do
In Exercises, find the indefinite integral. sin 2x cos 2x dx
The elliptic integralgives the circumference of an ellipse. Use Simpson's Rule with n = 8 to approximate the circumference. *π/2 8√3 7² 0 1-sin²0 de
In Exercises, find the indefinite integral. S. tan x sec² x dx
In Exercises, find the indefinite integral. x sin x² dx
Use the Trapezoidal Rule to estimate the number of square meters of land, where x and y are measured in meters, as shown in the figure. The land is bounded by a stream and two straight roads that meet at right angles X y 0 100 125 125 y X 600 700 95 88 150 Road 100- 50- 200 300 120 112 800 900 75
In Exercises, find the indefinite integral. csc² x cot³ x dx
To determine the size of the motor required to operate a press, a company must know the amount of work done when the press moves an object linearly 5 feet. The variable force to move the object iswhere F is given in pounds and x gives the position of the unit in feet. Use Simpson's Rule with n = 12
Finding an Equation In Exercises find an equation for the function ƒ that has the given derivative and whose graphpasses through the given point. Derivative f'(x) = - sin 2 Point (0, 6)
In Exercises, find the indefinite integral. fa sin x cos³ x dx
In Exercises use Simpson's Rule with n = 6 to approximate π using the given equation. -S π = TT 4 1 + x² dx
In Exercises use Simpson's Rule with n = 6 to approximate π using the givenequation. ㅠ= (1/2 0 6 √1 - x2 dx
Prove that Simpson's Rule is exact when approximating the integral of a cubic polynomial function, and demonstrate the result with n = 4 for So 0 x³ dx.
Finding an Equation In Exercises find an equation for the function ƒ that has the given derivative and whose graph passes through the given point. Derivative f'(x) = sec²(2x) Point 2
Use Simpson's Rule with n = 10 and a computer algebra system to approximate t in the integral equation S's 10 sin √√√x dx = 2.
Prove that you can find a polynomialthat passes through any three points (x1, y1), (x2, y2), and (X3, y3), where the xi's are distinct. p(x) = Ax² + Bx + C
Finding an Equation In Exercises find an equation for the function ƒ that has the given derivative and whose graph passes through the given point. Derivative f'(x) = 2x(4x² - 10)² Point (2, 10)
In Exercises find the indefinite integral by the method shown in Example 5.Data from in Example 5 [x√x + 6dx
Finding an Equation In Exercises find an equation for the function ƒ that has the given derivative and whose graph passes through the given point. Derivative f'(x) = 2x√ √8 - x² Point (2,7)
In Exercises find the indefinite integral by the method shown in Example 5.Data from in Example 5 SRVT- √1 - x dx
In Exercises find the indefinite integral by the method shown in Example 5.Data from in Example 5 fax + (x + 1)√2 - x dx
In Exercises find the indefinite integral by the method shown in Example 5.Data from in Example 5 Sav X 3x - 4 dx
In Exercises determine which value best approximates the definite integral. Make your selection on the basis of a sketch.(a) 5(b) -3(c) 10(d) 2(e) 8 So √x dx
In Exercises find the indefinite integral by the method shown in Example 5.Data from in Example 5 EXAMPLE 5 Rewriting Before Integrating X √x + 1² dx = [(5x + 7) dx S( = f(x1/2 -1/2 + x-1/2) dx x3/2 x1/2 3/2+1/2+ = 3x13³/2 + 2x1/² + C = + C 2 = ²/² √/x(x + 3) + C 3 Rewrite as two
In Exercises find the indefinite integral by the method shown in Example 5.Data from in Example 5 12 x² - 1 2x - 1 dx
In Exercises find the indefinite integral by the method shown in Example 5.Data from in Example 5 2x + 1 x + 4 dx
In Exercises evaluate the definite integral. Use a graphing utility to verify your result. So x√1 - x² dx
In Exercises find the indefinite integral by the method shown in Example 5.Data from in Example 5 Six -X (x + 1) - √√√x + 1 dx
In Exercises the graph of a function ƒ is shown. Use the differential equation and the given point to find an equation of the function. dy dx - 48 (3x + 5)³ f of 654 A (-1,3) +|||| + -6-5-4-3-2-1 12 -2+ X
In Exercises the graph of a function ƒ is shown. Use the differential equation and the given point to find an equation of the function. dx = 18x²(2x³ + 1)² -4-3-2 765 4 21 2 f (0,4) 1 2 3 4
In Exercises find the area of the region. Use a graphing utility to verify your result. *w/4 /w/12 y 4 3 2 1 csc 2x cot 2x dx 元 Зл 16 8 16 R100 4 X
In Exercises find the area of the region. Use a graphing utility to verify your result. [x 0 y 16 12 00 8 4 x √x + 1 dx 2 4 6 8
In Exercises find the area of the region. Use a graphing utility to verify your result. L -2 x² 3√x + 2 dx 80 60 40 20 -2 2 + 4 6 X
In Exercises find the area of the region. Use a graphing utility to verify your result. (2π/3 TT/2 y 4 3 2- sec² 4 1 3 dx I X
In Exercises evaluate the integral using the properties of even and odd functions as an aid. *π/2 J-π/2 sin² x cos x dx
In Exercises evaluate the integral using the properties of even and odd functions as an aid. J-2 x²(x² + 1) dx
In Exercises evaluate the integral using the properties of even and odd functions as an aid. x(x² + 1)³ dx J-2
Use the symmetry of the graphs of the sine and cosine functions as an aid in evaluating each definite integral.(a)(b)(c)(d) π/4 -π/4 sin x dx
In Exercises evaluate the integral using the properties of even and odd functions as an aid. *π/2 J-π/2 sin x cos x dx
Use ∫40 x2dx = 64/3 to evaluateeach definite integral without using the Fundamental Theoremof Calculus.(a)(b)(c)(d) 0 J-4 x² dx
Describe whywhere u = 5 - x². (5 x²)³ dx u³ du + [₁³ √x(5-x
In Exercises write the integral as the sum of the integral of an odd function and the integral of an even function. Use this simplification to evaluate the integral. S₁ -3 (x³ + 4x² 3x - 6) dx
You are asked to find one of the integrals. Which one would you choose? Explain.(a)(b) S √³+1 dx or [1²V x²√√x³ + 1 dx
In Exercises write the integral as the sum of the integral of an odd function and the integral of an even function. Use this simplification to evaluate the integral. *TT/2 J-TT/2 (sin 4x + cos 4x) dx
Find the indefinite integral in two ways. Explain any difference in the forms of the answers.(a)(b) S (2x - 1)² dx
Without integrating, explain why -2 x(x² + 1)² dx = 0.
The graph shows the flow rate of water at a pumping station for one day.(a) Approximate the maximum flow rate at the pumping station. At what time does this occur?(b) Explain how you can find the amount of water used during the day.(c) Approximate the two-hour period when the least amount of water
The oscillating current in an electrical circuit iswhere I is measured in amperes and t is measured in seconds. Find the average current for each time interval.(a)(b)(c) I = 2 sin(60πt) + cos(120mt)
Find by evaluating an appropriate definite integral over the interval [0, 1]. lim n→ +∞o i=1 sin(iπ/n) n
Rewriting Integrals(a) Show that(b) Show that So 0 = Lea 0 x²(1-x)5 dx = x5(1 x)² dx. -
The rate of depreciation dv/dt of a machine is inversely proportional to the square of (t + 1), where Vis the value of the machine t years after it was purchased. The initial value of the machine was $500,000, and its value decreased $100,000 in the first year. Estimate its value after 4 years.
In Exercises, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. [121 (2x + 1)² dx = (2x + 1)³ + C
In Exercises, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. |x(x² + 1) dx = x²(x³ + x) + C
In Exercises, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. 10 *10 Sto (ax³ + bx² + cx + d) dx = 2 -10 (bx² + d) dx
In Exercises, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. So sin x dx = cb+2# a sin x dx
In Exercises, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. 4 sin x cos x dx = cos 2x + C
In Exercises, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. sin² 2x cos 2x dx = sin³2x + C
Assume that ƒ is continuous everywhere and that c is a constant. Show that cb ca cb f(x) dx = c f(cx) dx. dx = cff f
Show that if ƒ is continuous on the entire real number line, then b [Fx a f(x + h) dx = = cb+h [ Ja+h f(x) dx.
Find all the continuous positive functions ƒ(x), for 0 ≤ x ≤ 1, such thatwhere a is a given real number. f ₁ f(x) dx = 1 Jo 6. f(x)x dx = a f(x)x² dx = a²
Exercises, solve the differential equation. dy dx || x + 1 (x² + 2x - 3)²
In Exercises complete the table by identifying u and du for the integral. f(g(x))g'(x) dx [ (8x² (8x² + 1)²(16x) dx u = g(x) du = g'(x) dx
In Exercises use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. JO TT 4 x² + 1 dx
In Exercises complete the table by identifying u and du for the integral. [ƒ (g(x))g'(x) dx [√x + 1dx u = g(x) du = g'(x) dx
In Exercises use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. TT cos x dx
In Exercises use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. -2 x√x² + 1 dx
In Exercises complete the table by identifying u and du for the integral. [f(g(x))g'(x) dx Stan² tan² x sec² x dx u = g(x) du = g'(x) dx
In Exercises complete the table by identifying u and du for the integral. [ƒ(g(x))g'(x) dx COS X dx sin² x u = g(x) du = g'(x) dx
In Exercises use a graphing utility to graph the integrand. Use the graph to determine whether the definite integral is positive, negative, or zero. LIV -2 x√2-x dx
In Exercises, find the indefinite integral and check the result by differentiation. √6²-5 (x² - 9)³(2x) dx
In Exercises, find the indefinite integral and check the result by differentiation. fa+ + 6x)*(6) dx
In Exercises evaluate the definite integral. Use a graphing utility to verify your result. L'₁ -3 8 dt
In Exercises evaluate the definite integral. Use a graphing utility to verify your result. 2 10 6x dx
In Exercises, find the indefinite integral and check the result by differentiation. √ √25 - x² (-2x) dx
In Exercises, find the indefinite integral and check the result by differentiation. 3/3 - 4x²(-8x) dx
In Exercises evaluate the definite integral. Use a graphing utility to verify your result. S. -1 (2x - 1) dx
In Exercises evaluate the definite integral. Use a graphing utility to verify your result. 1₁ - 1 (7 - 3t) dt
In Exercises, find the indefinite integral and check the result by differentiation. [x²³(x³ + 3)² dx
In Exercises, find the indefinite integral and check the result by differentiation. fre x²(6 - x³)³ dx
In Exercises evaluate the definite integral. Use a graphing utility to verify your result. L₁ (²- -1 (t²- 2) dt
In Exercises, find the indefinite integral and check the result by differentiation. x²(x² - 1)4 dx
In Exercises evaluate the definite integral. Use a graphing utility to verify your result. S² (6x² – 3x) dx
In Exercises evaluate the definite integral. Use a graphing utility to verify your result. So (2t 1)² dt
In Exercises, find the indefinite integral and check the result by differentiation. [x(5x² + 4)³ dx
In Exercises evaluate the definite integral. Use a graphing utility to verify your result. S² (4x³ - 3x²) dx
In Exercises, find the indefinite integral and check the result by differentiation. #PZ + 1/
In Exercises evaluate the definite integral. Use a graphing utility to verify your result. S₁² [² (21/2 - 1) dx
In Exercises, find the indefinite integral and check the result by differentiation. 5x 3/1-x² dx
In Exercises, find the indefinite integral and check the result by differentiation. 3√√214 + 3 dt
In Exercises evaluate the definite integral. Use a graphing utility to verify your result. u 2 u du
In Exercises evaluate the definite integral. Use a graphing utility to verify your result. du ( ² - ₂ ) J n
In Exercises, find the indefinite integral and check the result by differentiation. [11² √ u³ + 2 du
In Exercises evaluate the definite integral. Use a graphing utility to verify your result. J-8 1/3 X dx
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