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mathematics
calculus 10th edition
Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions
Two circles of radius 3, with centers at (-2, 0) and (2, 0), intersect as shown in the figure. Find the area of the shaded region.Evaluate Jo In(x 2 x² + 1 + 1) dx.
Show that the arc length of the graph of y = sin x on the interval [0, 2π] is equal to the circumference of the ellipse x² + 2y² = 2 (see figure). Зл 2 П П 2 -П П 2л X
Two circles of radius 3, with centers at (-2, 0) and (2, 0), intersect as shown in the figure. Find the area of the shaded region. 4 + -6 -4-3-2 li 2 -4- H 2 3 4 6
Use trigonometric substitution to verify the integration formulas given in Theorem 8.2.Data from in Theorem 8.2 THEOREM 8.2 Special Integration Formulas (a > 0) •√√₁² Sv √u². 1. 2. ·S. 3. a² − u² du - И = =(₁² a² arcsin + + a = 1/2/(₁+√/u² a² du = u√a² u² + a² du
In Exercises find the area of the region bounded by the graphs of the equations. y = cos² x, y = sin x cos x, X = TT 2' - x= 4
The crescent-shaped region bounded by two circles forms a lune (see figure). Find the area of the lune given that the radius of the smaller circle is 3 and the radius of the larger circle is 5. 3 5
In Exercises find the area of the region bounded by the graphs of the equations. πT y sin x, y = sin³x, x=0, x= 2
In Exercises find the area of the region bounded by the graphs of the equations. y sin² mx, y = 0, x=0, x = 1
Evaluate the following two integrals, which yield the fluid forces given below.(a)(b) Finside 48 0.8 1 (0.8y)(2)√1- y² dy
In Exercises(a) Find the indefinite integral in two different ways. (b) Use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant. (c) Verify analytically that the results differ only
Use the graph of ƒ'shown in the figure to answer the following.(a) Using the interval shown in the graph, approximate the value(s) of x where ƒ is maximum. Explain. (b) Using the interval shown in the graph, approximate the value(s) of x where ƒ is minimum. Explain.
Find the fluid force on a circular observation window of radius 1 foot in a vertical wall of a large water-filled tank at a fish hatchery when the center of the window is (a) 3 feet and (b) d feet(d > 1) below the water'ssurface (see figure). Usetrigonometric substitutionto evaluate
In Exercises(a) Find the indefinite integral in two different ways. (b) Use a graphing utility to graph the antiderivative (without the constant of integration) obtained by each method to show that the results differ only by a constant. (c) Verify analytically that the results differ only
The field strength H of a magnet of length 2L on a particle r units from the center of the magnet iswhere ±m are the poles of the magnet (see figure). Find the average field strength as the particle moves from 0 to R units from the center by evaluating the integral H = 2mL (r² + [²)3/2
Find the surface area of the solid generated by revolving the region bounded by the graphs of y = x², y = 0, x = 0, and x = √2 about the x-axis.
Evaluate ∫ sin x cos x dx using the given method. Explain how your answers differ for each method.(a) Substitution where u = sin x(b) Substitution where u = cos x(c) Integration by parts(d) Using the identity sin 2x = 2 sin x cos x
In Exercises find the centroid of the region determined by the graphs of the inequalities. y ≤ x², (x-4)² + y² ≤ 16, y ≥ 0
In Exercises find the centroid of the region determined by the graphs of the inequalities. y ≤ 3/√√x² + 9₁ y ≥ 0, x ≥ 4, x ≤ 4
In your own words, describe how you would integrate ∫ secm x tann x dx for each condition.(a) m is positive and even. (b) n is positive and odd.(c) n is positive and even, and there are no secant factors.(d) m is positive and odd, and there are no tangent factors.
In Exercises evaluate the definite integral. TT/2 J-T/2 (sin²x + 1) dx
In your own words, describe how you would integrate ∫ sinm x cosn x dx for each condition.(a) m is positive and odd. (b) n is positive and odd.(c) m and n are both positive and even.
In Exercises evaluate the definite integral. TT/2 J-T/2 3 cos³ x dx
(a) Find formulas for the distances between (0, 0) and (a, a²) along the line between these points and along the parabola y = x².(b) Use the formulas from part (a) to find the distances for a = 1 and a = 10.(c) Make a conjecture about the difference between the twodistances as a increases.
Show that the length of one arch of the sine curve is equal to the length of one arch of the cosine curve.
In Exercises evaluate the definite integral. *π/3 Jπ/6 sin 6x cos 4x dx
In Exercises find the arc length of the curve over the given interval. y = x², [0, 4]
In Exercises find the arc length of the curve over the given interval. y = In x, [1, 5]
In Exercises evaluate the definite integral. TT/3 tan² x dx
In Exercises evaluate the definite integral. TT/2 0 cos t 1 + sin t dt
In Exercises find the volume of the torus generated by revolving the region bounded by the graph of the circle about the y-axis. (xh)² + y² = r², h>r
In Exercises find the volume of the torus generated by revolving the region bounded by the graph of the circle about the y-axis. (x - 3)² + y² = 1
In Exercises evaluate the definite integral. π/4 So 0 6 tan³ x dx
The axis of a storage tank in the form of a right circular cylinder is horizontal (see figure). The radius and length of the tank are 1 meter and 3 meters, respectively.(a) Determine the volume of fluid in the tank as a function of its depth d.(b) Use a graphing utility to graph the function in
The surface of a machine part is the region between the graphs of y = |x| and x² + (y - k)² = 25 (see figure).(a) Find k when the circle is tangent to the graph of y = |x|.(b) Find the area of the surface of the machine part.(c) Find the area of the surface of the machine part as afunction of the
Find the area of the shaded region of the circle of radius a when the chord is h units (0 < h < a) from the center of the circle (see figure). X D 41 -a D X D-
In Exercises find the indefinite integral. Use a computer algebra system to confirm your result. 1 - sec t cost - 1 dt
In Exercises find the indefinite integral. Use a computer algebra system to confirm your result. 1 - sec t cost - 1 dt
Find the area enclosed by the ellipse shown in the figure. x2 X a + 6 = 1
In Exercises find the indefinite integral. Use a computer algebra system to confirm your result. (tant sect) dt -
In Exercises find the indefinite integral. Use a computer algebra system to confirm your result. sin² x - cos² x COS 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. If x= sin 0, then L₁Rv π/2 dx = 2 ["² x² √/1 − x² dx = 2 - sin²0 cos² 0 do.
In Exercises find the indefinite integral. Use a computer algebra system to confirm your result. 1 sec x tan x - dx
In Exercises find the indefinite integral. Use a computer algebra system to confirm your result. sin² x - cos² x COS 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. If x= sec 0, then 2 S [√x² = 1 dx = [sec X sec 0 tan 0 do.
In Exercises find the indefinite integral. Use a computer algebra system to confirm your result. cot³ t csc t dt
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. If x= tan 0, then Love dx (1 + x²)³/2 (4π/3 cos 0 de.
In Exercises find the indefinite integral. Use a computer algebra system to confirm your result. cot² t CSC t dt
In Exercises find the indefinite integral. Use a computer algebra system to confirm your result. X X cot³csc4dx 2 2
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. If x= sin 0, then S S dx √√1-x² || do.
(a) Find the integralusing u-substitution.Then find the integral using trigonometric substitution. Discuss the results.(b) Find the integralalgebraically usingThen find the integral using trigonometric substitution. Discuss the results. X -2 X x² + 9 dx
In Exercises find the indefinite integral. Use a computer algebra system to confirm your result. So csc4 3x dx
Use the graph of ƒ' shown in the figure to answer the following.(a) Identify the open interval(s) on which the graph of ƒ is increasing or decreasing. Explain. (b) Identify the open interval(s) on which the graph of ƒ is concave upward or concave downward. Explain. 4 2 y f'(x)
In Exercises find the indefinite integral. Use a computer algebra system to confirm your result. 1₁ X X tan sec4 dx 4
State the substitution you would make if you used trigonometric substitution for an integral involving the given radical, where a > 0. Explain your reasoning.(a)(b)(c) 2 u² 2
State the method of integration you would use to perform each integration. Explain why you chose that method. Do not integrate.(a)(b) five x√x² + 1 dx
In Exercises find the indefinite integral. Use a computer algebra system to confirm your result. So cot³ 2x dx
In Exercises evaluate the definite integral using (a) The given integration limits (b) The limits obtained by trigonometric substitution 6 J4 x² √x²-9 dx
In Exercises evaluate the definite integral using (a) The given integration limits (b) The limits obtained by trigonometric substitution 3/5 0 /9 - 25x² dx
In Exercises evaluate the definite integral using (a) The given integration limits (b) The limits obtained by trigonometric substitution 8. 9I - zx/ J4 x2 16 dx
In Exercises evaluate the definite integral using (a) The given integration limits (b) The limits obtained by trigonometric substitution X3 x² + 9 = dx
In Exercises evaluate the definite integral using (a) The given integration limits (b) The limits obtained by trigonometric substitution √3/2 10 1 (1 - 1²) 5/2 dt
In Exercises evaluate the definite integral using (a) The given integration limits (b) The limits obtained by trigonometric substitution 10 √3/2 1² (112) 3/2 dt
In Exercises complete the square and find the indefinite integral. S= x² 2x - x2 dx x²
In Exercises complete the square and find the indefinite integral. -2 X 6x + 5 dx
In Exercises complete the square and find the indefinite integral. X x² + 6x + 12 2 dx
In Exercises find the indefinite integral. S arcsec 2x dx, x 1 2
In Exercises complete the square and find the indefinite integral. 1 4x – x2 dx
In Exercises find the indefinite integral. [ + x + EX x4 + 2x² + 1 dx
In Exercises find the indefinite integral. 1 4 + 4x² + x4 2 dx
In Exercises find the indefinite integral. x arcsin x dx
In Exercises find the indefinite integral. 1- x √x dx
In Exercises find the indefinite integral. 1 X x√ √9x² + 1 dx
In Exercises find the indefinite integral. Sex√T ex√√1-e²x dx
In Exercises find the indefinite integral. 1 (x² + 5)3/2 dx
In Exercises find the indefinite integral. S 25x² + 4 XA dx
In Exercises find the indefinite integral. - 3x (x² + 3)3/2 dx
In Exercises find the indefinite integral. S 1 x√ √4x² + 9 X dx
In Exercises find the indefinite integral. S. x² 36x²2 dx
In Exercises find the indefinite integral. 1-x² x4 dx
In Exercises find the indefinite integral. 1 x² - 4 dx
In Exercises find the indefinite integral. s √16 - 4x² dx
In Exercises find the indefinite integral using the substitution x = tan θ. x² (1 + x²)2 dx
In Exercises find the indefinite integral. 1 16x2 dx
In Exercises use the Special Integration Formulas (Theorem 8.2) to find the indefinite integral.Data from in Theorem 8.2 THEOREM 8.2 Special Integration Formulas (a > 0) 1. 1. S√α² - 1² du ² = = 1/(₁²³= 2. 3. /u²2 - a² du Sv √u² + a² du a arcsin a + uv@ -ử +c = = 1/²
In Exercises find the indefinite integral using the substitution x = tan θ. 1 (1 + x²)² ¡dx
In Exercises use the Special Integration Formulas (Theorem 8.2) to find the indefinite integral.Data from in Theorem 8.2 THEOREM 8.2 Special Integration Formulas (a > 0) 1. 1. S√α² - 1² du ² = = 1/(₁²³= 2. 3. /u²2 - a² du Sv √u² + a² du a arcsin a + uv@ -ử +c = = 1/²
In Exercises find the indefinite integral using the substitution x = tan θ. 9x³ √1 + x² dx
In Exercises use the Special Integration Formulas (Theorem 8.2) to find the indefinite integral.Data from in Theorem 8.2 THEOREM 8.2 Special Integration Formulas (a > 0) 1. 1. S√α² - 1² du ² = = 1/(₁²³= 2. 3. /u²2 - a² du Sv √u² + a² du a arcsin a + uv@ -ử +c = = 1/²
In Exercises find the indefinite integral using the substitution x = 5 sec θ. x³ x²-25 dx
In Exercises find the indefinite integral using the substitution x = 5 sec θ. x² - X 25 - dx
In Exercises find the indefinite integral using the substitution x = tan θ. x√1 + x² dx
In Exercises find the indefinite integral using the substitution x = 4 sin θ. x³ √16x² dx
In Exercises find the indefinite integral using the substitution x = 5 sec θ. SPVR- x³√x² - 25 dx
In Exercises find the indefinite integral using the substitution x = 4 sin θ. 4 x²√√16x² zdx
In Exercises find the indefinite integral using the substitution x = 4 sin θ. 1 (16- x²)3/2 dx
In Exercises find the indefinite integral using the substitution x = 4 sin θ. √16x² X - dx
In Exercises state the trigonometric substitution you would use to find the indefinite integral. Do not integrate. [x²(x² - 25)3/2 dx
In Exercises state the trigonometric substitution you would use to find the indefinite integral. Do not integrate. /25x² dx
In Exercises state the trigonometric substitution you would use to find the indefinite integral. Do not integrate. S √4x² dx
In Exercises state the trigonometric substitution you would use to find the indefinite integral. Do not integrate. Je (9 + x²)-² dx
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