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6th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781285774770

Chapter 8.6, Problem 72E

Textbook Problem

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Building DesignThe cross section of a precast concrete beam for a building is bounded by the graphs of the equations

where *x* and *y* are measured in feet. The length of the beam is 20 feet (see figure).

(a) Find the volume *V* and the weight *W* of the beam. Assume the concrete weighs 148 pounds per cubic foot.

(b) Find the centroid of a cross section of the beam.

**(a)**

To determine

**To calculate:** The cross section of the precast concrete beam of the building that is bounded by these equations,

**Given:** The equations for the beams are:

The length of the beam is

**Formula used:**

Volume of the beam:

And

**Calculation:**

Since, the length of the beam is

Now, find the volume of the beam

**(b)**

To determine

The centroid of the cross section of the concrete beam.

Calculus: Early Transcendental Functions (MindTap Course List)

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Use the graph of f shown in the...Ch. 8.3 - sec43xtan3xdxCh. 8.3 - Comparing Methods In Exercises 69 and 70, (a) find...Ch. 8.3 - Area In Exercises 73 and 74, find the area of the...Ch. 8.3 - Area In Exercises 7174, find the area of the...Ch. 8.3 - Area In Exercises 7174, find the area of the...Ch. 8.3 - Area In Exercises 73 and 74, find the area of the...Ch. 8.3 - Volume In Exercises 75 and 76, find the volume of...Ch. 8.3 - Volume In Exercises 75 and 76, find the volume of...Ch. 8.3 - Volume and Centroid In Exercises 77 and 78, for...Ch. 8.3 - Volume and Centroid In Exercises 77 and 78, for...Ch. 8.3 - Verifying a Reduction Formula In Exercises 79-82,...Ch. 8.3 - Verifying a Reduction Formula In Exercises 79-82,...Ch. 8.3 - Verifying a Reduction Formula In Exercises 79-82,...Ch. 8.3 - Verifying a Reduction Formula In Exercises 79-82,...Ch. 8.3 - Using Formulas In Exercises 83-86, Find the...Ch. 8.3 - Using Formulas In Exercises 85-86, find the...Ch. 8.3 - Using Formulas In Exercises 83-86, find the...Ch. 8.3 - Verifying a Reduction FormulaIn Exercises 7982,...Ch. 8.3 - Modeling DataThe table shows the normal maximum...Ch. 8.3 - Walliss Formulas Use the result of Exercise 80 to...Ch. 8.3 - Orthogonal Functions The inner product of two...Ch. 8.3 - Fourier Series The following sum is a finite...Ch. 8.4 - Trigonometric Substitution In Exercises 14, state...Ch. 8.4 - Trigonometric Substitution In Exercises 14, state...Ch. 8.4 - Trigonometric Substitution In Exercises 14, state...Ch. 8.4 - Trigonometric Substitution In Exercises 14, state...Ch. 8.4 - Using trigonometric Substitution In Exercises 3-6,...Ch. 8.4 - Using trigonometric Substitution In Exercises 3-6,...Ch. 8.4 - Using trigonometric Substitution In Exercises 3-6,...Ch. 8.4 - Using trigonometric Substitution In Exercises 3-6,...Ch. 8.4 - Using Trigonometric Substitution In Exercises...Ch. 8.4 - Using Trigonometric Substitution In Exercises...Ch. 8.4 - Using Trigonometric Substitution In Exercises...Ch. 8.4 - Using Trigonometric Substitution In Exercises...Ch. 8.4 - Special Integration Formulas In Exercises 15-18,...Ch. 8.4 - Using Trigonometric Substitution In Exercises...Ch. 8.4 - Using Trigonometric Substitution In Exercises...Ch. 8.4 - Using Trigonometric Substitution In Exercises...Ch. 8.4 - Using Trigonometric Substitution In Exercises...Ch. 8.4 - Using Formulas In Exercises 1720, use the Special...Ch. 8.4 - Using Formulas In Exercises 1720, use the Special...Ch. 8.4 - Using FormulasIn Exercises 1720, use the Special...Ch. 8.4 - Finding an Indefinite Integral In Exercises 2136,...Ch. 8.4 - Finding an Indefinite IntegralInExercises 2136,...Ch. 8.4 - Finding an Indefinite Integral In Exercises 19-32,...Ch. 8.4 - Finding an Indefinite Integral In Exercises 19-32,...Ch. 8.4 - Finding an Indefinite Integral In Exercises 19-32,...Ch. 8.4 - Finding an Indefinite Integral In Exercises19-32,...Ch. 8.4 - Finding an Indefinite Integral In Exercises19-32,...Ch. 8.4 - Finding an Indefinite Integral In Exercises19-32,...Ch. 8.4 - Finding an Indefinite Integral In Exercises19-32,...Ch. 8.4 - Finding an Indefinite Integral In Exercises19-32,...Ch. 8.4 - Finding an Indefinite Integral In Exercises19-32,...Ch. 8.4 - Finding an Indefinite Integral In Exercises19-32,...Ch. 8.4 - Finding an Indefinite Integral In Exercises19-32,...Ch. 8.4 - Finding an Indefinite Integral In Exercises19-32,...Ch. 8.4 - Finding an Indefinite Integral In Exercises 19-32,...Ch. 8.4 - Finding an Indefinite Integral In Exercises 19-32,...Ch. 8.4 - Completing the Square In Exercises 33-36, find the...Ch. 8.4 - Completing the Square In Exercises 33-36, complete...Ch. 8.4 - Completing the Square In Exercises 33-36, find the...Ch. 8.4 - Completing the Square In Exercises 33-36, find the...Ch. 8.4 - Converting the Limits of Integration In Exercises...Ch. 8.4 - Converting the Limits of Integration In Exercises...Ch. 8.4 - Converting the Limits of Integration In Exercises...Ch. 8.4 - Converting the Limits of Integration In Exercises...Ch. 8.4 - Converting the Limits of Integration In Exercises...Ch. 8.4 - Converting the Limits of Integration In Exercises...Ch. 8.4 - Trigonometric SubstitutionState the substitution...Ch. 8.4 - Choosing a MethodState the method of integration...Ch. 8.4 - Comparing Methods (a) Find the integral x1x2dx dx...Ch. 8.4 - How do you see it? Use the graph of f1 shown in...Ch. 8.4 - True or False? In Exercises 47-50, determine...Ch. 8.4 - True or False? In Exercises 47-50, determine...Ch. 8.4 - True or False? In Exercise 47-50 Determine whether...Ch. 8.4 - True or False? In Exercise 47-50 Determine whether...Ch. 8.4 - Area Find the Area enclosed by the ellipse...Ch. 8.4 - Area Find the area of the shaded region of the...Ch. 8.4 - Arc Length In Exercises 53 and 54, find the arc...Ch. 8.4 - Volume of a Torus In Exercises 55 and 56, find the...Ch. 8.4 - Volume of a Torus In Exercises 55 and 56, find the...Ch. 8.4 - Centroid In Exercises 57 and 58, find the centroid...Ch. 8.4 - Centroid In Exercise 57 and 58, Find the Centroid...Ch. 8.4 - Mechanical Design The surface of a machine part is...Ch. 8.4 - Volume The axis of a storage tank in the form of a...Ch. 8.4 - Field Strength The field strength H of a magnet of...Ch. 8.4 - Arc LengthIn Exercises 61 and 62, find the arc...Ch. 8.4 - Arc Length Show that the length of one arch of the...Ch. 8.4 - Conjecture (a) Find formulas for the distances...Ch. 8.4 - Fluid Force Find the fluid force on a circular...Ch. 8.4 - Surface AreaFind the surface area of the solid...Ch. 8.4 - Fluid Force Evaluate the following two integrals,...Ch. 8.4 - Verifying Formulas Use trigonometric substitution...Ch. 8.4 - Arc length show that the arc length of the graph...Ch. 8.4 - Area of a Lune The crescent shaped region bounded...Ch. 8.4 - Area: Two circles of radius 3, with centres at...Ch. 8.4 - Evaluate 01ln(x+1)x2+1dxCh. 8.5 - Partial Fraction Decomposition In Exercises 14,...Ch. 8.5 - Partial Fraction Decomposition In Exercises 14,...Ch. 8.5 - Partial Fraction Decomposition In Exercises 14,...Ch. 8.5 - Partial Fraction Decomposition In Exercises 14,...Ch. 8.5 - Using Partial Fractions In Exercises 3-20, use...Ch. 8.5 - Using Partial Fractions In Exercises 3-20, use...Ch. 8.5 - Using Partial Fractions In Exercises 3-20, use...Ch. 8.5 - Using Partial Fractions In Exercises 3-20, use...Ch. 8.5 - Using Partial Fractions In Exercises 3-20, use...Ch. 8.5 - Using Partial Fractions In Exercises 3-20, use...Ch. 8.5 - Using Partial Fractions In Exercises 3-20, use...Ch. 8.5 - Using Partial Fractions In Exercises 3-20, use...Ch. 8.5 - Using Partial Fractions In Exercises 3-20, use...Ch. 8.5 - Using Partial Fractions In Exercises 3-20, use...Ch. 8.5 - Using Partial Fractions In Exercises 522, use...Ch. 8.5 - Using Partial Fractions In Exercises 3-20, use...Ch. 8.5 - Using Partial Fractions In Exercises 522, use...Ch. 8.5 - Using Partial Fractions In Exercises 3-20, use...Ch. 8.5 - Using partial Fractions In Exercises 3-20, use...Ch. 8.5 - Using Partial Fractions In Exercises 3-20, use...Ch. 8.5 - Using partial Fractions In Exercises 3-20, use...Ch. 8.5 - Using partial Fractions In Exercises 3-20, use...Ch. 8.5 - Evaluating a Definite Integral In Exercises 2124,...Ch. 8.5 - Evaluating a Definite Integral In Exercises 21-24,...Ch. 8.5 - Evaluating a Definite Integral In Exercises 21-24,...Ch. 8.5 - Evaluating a Definite Integral In exercises 21-24,...Ch. 8.5 - Finding an Indefinite Integral In Exercises 25-32,...Ch. 8.5 - Finding an Indefinite Integral In Exercises 25-32,...Ch. 8.5 - Finding an Indefinite Integral In Exercises 25-32,...Ch. 8.5 - Finding an Indefinite Integral In Exercises 25-32,...Ch. 8.5 - Finding an Indefinite Integral In Exercises 25-32,...Ch. 8.5 - Finding an Indefinite Integral In Exercises 25-32,...Ch. 8.5 - Finding an Indefinite Integral In Exercises 25-32,...Ch. 8.5 - Finding an Indefinite Integral In Exercises 25-32,...Ch. 8.5 - Verifying a Formula In Exercises 33-36, use the...Ch. 8.5 - Verifying a Formula In Exercises 33-36, use the...Ch. 8.5 - Verifying a Formula In Exercises 33-36, use the...Ch. 8.5 - Verifying a Formula In Exercises 33-36, use the...Ch. 8.5 - Using Partial FractionsWhat is the first step when...Ch. 8.5 - DecompositionDescribe the decomposition of the...Ch. 8.5 - Choosing a MethodState the method you would use to...Ch. 8.5 - HOW DO YOU SEE IT? Use the graph of f1 shown in...Ch. 8.5 - Area Find the area of the region bounded by the...Ch. 8.5 - Area Find the area of the region bounded by the...Ch. 8.5 - Modeling Data The predicted cost C (in hundreds of...Ch. 8.5 - Logistic Growth In Chapter 6, the exponential...Ch. 8.5 - Volume and Centroid Consider the region bounded by...Ch. 8.5 - Volume Consider the region bounded by the graph of...Ch. 8.5 - Epidemic Model A single infected individual enters...Ch. 8.5 - Chemical Reaction In a chemical reaction, one unit...Ch. 8.5 - Using Two Methods Evaluce 01x1+x4dx in two...Ch. 8.5 - Prove 227=01x4(1x)41+x2dxCh. 8.6 - Integration by Tables In Exercises 1 and 2, use a...Ch. 8.6 - Integration by Tables In Exercises 1 and 2, use a...Ch. 8.6 - Integration by Tables In Exercises 3 and 4, use a...Ch. 8.6 - Integration by Tables In Exercises 3 and 4, use a...Ch. 8.6 - Integration by Tables In Exercises 58, use a table...Ch. 8.6 - Integration by Tables In Exercises 58, use a table...Ch. 8.6 - Integration by Tables In Exercises 58, use a table...Ch. 8.6 - Integration by Tables In Exercises 58, use a table...Ch. 8.6 - Integration by Tables In Exercises 9 and 10, use a...Ch. 8.6 - Integration by Tables In Exercises 9 and 10, use a...Ch. 8.6 - Integration by Tables In Exercises 11 and 12, use...Ch. 8.6 - Integration by Tables In Exercises 11 and 12, use...Ch. 8.6 - Using Two Methods In Exercises 1316, find the...Ch. 8.6 - Using Two Methods In Exercises 1516, find the...Ch. 8.6 - Using Two Methods In Exercises 1316, find the...Ch. 8.6 - Using Two Methods In Exercises 1516, find the...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite IntegralIn Exercises...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite IntegralIn Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - ex(1e2x)3/2dxCh. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Finding an Indefinite Integral In Exercises 1738,...Ch. 8.6 - Evaluating a Definite Integral In Exercises 3946,...Ch. 8.6 - Evaluating a Definite Integral In Exercises 3946,...Ch. 8.6 - Evaluating a Definite Integral In Exercises 3946,...Ch. 8.6 - Evaluating a Definite Integral In Exercises 3946,...Ch. 8.6 - Evaluating a Definite Integral In Exercises 3946,...Ch. 8.6 - Evaluating a Definite Integral In Exercises 3946,...Ch. 8.6 - Evaluating a Definite Integral In Exercises 3946,...Ch. 8.6 - Evaluating a Definite Integral In Exercises 3946,...Ch. 8.6 - Verifying a Formula In Exercises 4752, verify the...Ch. 8.6 - Verifying a Formula In Exercises 4752, verify the...Ch. 8.6 - Verifying a Formula In Exercises 4752, verify the...Ch. 8.6 - Verifying a Formula In Exercises 4752, verify the...Ch. 8.6 - Verifying a Formula In Exercises 4752, verify the...Ch. 8.6 - Verifying a Formula In Exercises 4752, verify the...Ch. 8.6 - Finding or Evaluating an Integral In Exercises...Ch. 8.6 - Finding or Evaluating an Integral In Exercises...Ch. 8.6 - Finding or Evaluating an Integral In Exercises...Ch. 8.6 - Finding or Evaluating an Integral In Exercises...Ch. 8.6 - Finding or Evaluating an Integral In Exercises...Ch. 8.6 - Finding or Evaluating an Integral In Exercises...Ch. 8.6 - Finding or Evaluating an Integral In Exercises...Ch. 8.6 - Finding or Evaluating an Integral In Exercises...Ch. 8.6 - Area In Exercises 61 and 62, find the area of the...Ch. 8.6 - Area In Exercises 61 and 62, find the area of the...Ch. 8.6 - Finding a Pattern (a) Evaluate xn In x dx for n =...Ch. 8.6 - Reduction Formula Describe what is meant by a...Ch. 8.6 - Choosing a MethodState (if possible) the method or...Ch. 8.6 - HOW DO YOU SEE IT?Use the graph of f' shown in the...Ch. 8.6 - True or False? In Exercises 67 and 68, determine...Ch. 8.6 - True or False? In Exercises 67 and 68, determine...Ch. 8.6 - WorkA hydraulic cylinder on an industrial machine...Ch. 8.6 - Work Repeat Exercise 69, using F(x) =...Ch. 8.6 - VolumeConsider the region bounded by the graphs of...Ch. 8.6 - Building DesignThe cross section of a precast...Ch. 8.6 - Population A population is growing according to...Ch. 8.6 - PUTNAM EXAM CHALLENGE Evaluate0/2dx1+(tanx)2. 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In Exercises 9598, determine...Ch. 8.7 - True or False? In Exercises 9598, determine...Ch. 8.7 - True or False? In Exercises 9598, determine...Ch. 8.7 - True or False? 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Convergence or Divergence In Exercises 5362, use...Ch. 8.8 - Convergence or Divergence In Exercises 53-60, use...Ch. 8.8 - Convergence or Divergence In Exercises 53-60, use...Ch. 8.8 - Convergence or Divergence In Exercises 53-60, use...Ch. 8.8 - Convergence or Divergence In Exercises 53-60, use...Ch. 8.8 - Convergence or Divergence In Exercises 5362, use...Ch. 8.8 - Convergence or Divergence In Exercises 5362, use...Ch. 8.8 - Convergence or Divergence In Exercises 53-60, use...Ch. 8.8 - Convergence or Divergence In Exercises 53-60, use...Ch. 8.8 - Improper IntegralsDescribe the different types of...Ch. 8.8 - Improper Integrals Define the terms converges and...Ch. 8.8 - EXPLORING CONCEPTS Improper IntegralExplain why...Ch. 8.8 - EXPLORING CONCEPTS Improper Integral Consider the...Ch. 8.8 - Area In Exercises 63-66, find the area of...Ch. 8.8 - Area In Exercises 63-66, find the area of...Ch. 8.8 - Area In Exercises 63-66, find the area of...Ch. 8.8 - Area In Exercises 63-66, find the area of...Ch. 8.8 - 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Laplace...Ch. 8.8 - The Gamma Function The Gamma Function (n)...Ch. 8.8 - Proof Prove that In=(n1n+2)In1, where...Ch. 8.8 - Finding a Value For what value of e does the...Ch. 8.8 - Finding a Value For what value of c does the...Ch. 8.8 - Volume Find the volume of the solid generated by...Ch. 8.8 - Volume Find the volume of the solid generated...Ch. 8.8 - u-Substitution In Exercises 105 and 106, rewrite...Ch. 8.8 - u-Substitution In Exercises 105 and 106, rewrite...Ch. 8.8 - Rewriting an Integral Let f(x)dx be convergent and...Ch. 8 - xx236dxCh. 8 - xex21dxCh. 8 - xx249dxCh. 8 - Using Basic Integration Rules In Exercises 1-8,...Ch. 8 - Using Basic Integration Rules In Exercises 1-8,...Ch. 8 - Using Basic Integration Rules In Exercises 1-8,...Ch. 8 - Using Basic Integration Rules In Exercises 1-8,...Ch. 8 - Using Basic Integration Rules In Exercises 1-8,...Ch. 8 - xe3xdxCh. 8 - x3exdxCh. 8 - Using Integration by Parts In Exercises 9-16, use...Ch. 8 - Using Integration by Parts In Exercises 9-16, use...Ch. 8 - 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Using Different Methods In Exercises 35 and 36,...Ch. 8 - x39x2x12dx.Ch. 8 - Using Partial Fractions In Exercises 37-44, use...Ch. 8 - Using Partial Fractions In Exercises 37-44, use...Ch. 8 - Using Partial Fractions In Exercises 37-44, use...Ch. 8 - Using Partial FractionsIn Exercises 3338, use...Ch. 8 - Using Partial Fractions In Exercises 37-44, use...Ch. 8 - Integration by Tables In Exercises 49-56, use...Ch. 8 - Integration by Tables In Exercises 49-56, use...Ch. 8 - Integration by Tables In Exercises 49-56, use...Ch. 8 - Integration by Tables In Exercises 49-56, use...Ch. 8 - Integration by Tables In Exercises 49-56, use...Ch. 8 - Integration by Tables In Exercises 49-56, use...Ch. 8 - Integration by Tables In Exercises 49-56, use...Ch. 8 - Integration by Tables In Exercises 49-56, use...Ch. 8 - Verifying a FormulaVerify the reduction formula...Ch. 8 - Verifying a FormulaVerify the reduction formula...Ch. 8 - Finding an Indefinite Integral In Exercises 57-64,...Ch. 8 - Finding an Indefinite Integral In Exercises 57-64,...Ch. 8 - 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Area In Exercises 75 and 76, find the area of the...Ch. 8 - Centroid In Exercises 77 and 78, find the centroid...Ch. 8 - Centroid In Exercises 77 and 78, find the centroid...Ch. 8 - Arc Length In Exercises 71 and 72, approximate to...Ch. 8 - Arc Length In Exercises 71 and 72, approximate to...Ch. 8 - limx1(lnx)2x1Ch. 8 - Evaluating a Limit In Exercises 7380, use LHpitals...Ch. 8 - Evaluating a Limit In Exercises 7380, use LHpitals...Ch. 8 - Evaluating a Limit In Exercises 7380, use LHpitals...Ch. 8 - Evaluating a Limit In Exercises 7380, use LHpitals...Ch. 8 - Evaluating a Limit In Exercises 7380, use LHpitals...Ch. 8 - Evaluating a Limit In Exercises 7380, use LHpitals...Ch. 8 - Evaluating a Limit In Exercises 7380, use LHpitals...Ch. 8 - Evaluating an Improper Integral In Exercises...Ch. 8 - Evaluating an Improper Integral In Exercises...Ch. 8 - Evaluating an Improper Integral In Exercises...Ch. 8 - Evaluating an Improper Integral In Exercises...Ch. 8 - Evaluating an Improper Integral In Exercises...Ch. 8 - Evaluating an Improper Integral In Exercises...Ch. 8 - Evaluating an Improper Integral In Exercises...Ch. 8 - Evaluating an Improper Integral In Exercises...Ch. 8 - Present Value The board of directors of a...Ch. 8 - Volume Find the volume of the solid generated by...Ch. 8 - Probability The average lengths (from beak to...Ch. 8 - Walliss Formulas (a) Evaluate the integrals...Ch. 8 - Proof (a) Evaluate the integrals...Ch. 8 - Finding a Value Find the value of the positive...Ch. 8 - Finding a Value Find the value of the positive...Ch. 8 - LengthThe line x = 1 is tangent to the unit circle...Ch. 8 - Finding a LimitThe segment BD is the height of...Ch. 8 - Area Consider the problem of finding the area of...Ch. 8 - Area Use the substitution u=tanx2 to find the area...Ch. 8 - Arc Length Find the arc length of the graph of the...Ch. 8 - Centroid Find the centroid of the region bounded...Ch. 8 - Finding Limits Use a graphing utility to estimate...Ch. 8 - Inverse Function and Area (a) Let y=f1(x) be the...Ch. 8 - Area Factor the polynomial p(x)=x4+1 and then find...Ch. 8 - Partial Fraction Decomposition Suppose the...Ch. 8 - Partial Fraction Decomposition Use the result of...Ch. 8 - Evaluating an Integral (a) Use the substitution...Ch. 8 - Elementary Functions Some elementary functions,...Ch. 8 - Rocket The velocity v (in feet per second) of a...Ch. 8 - Proof Suppose that f(a)=f(b)=g(a)=g(b)=0 and the...Ch. 8 - Proof Suppose that f(a)=f(b)=0 and the second...Ch. 8 - Approximating an Integral Using the inequality...Ch. 8 - Volume Consider die shaded region between die...

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