application of integration exercises

4) [T] Under the curve of \( y=3x,\) \(x=0,\) and \( x=3\) rotated around the \(x\)-axis. 0 b= 4πa2b/3. For exercises 41 - 45, draw the region bounded by the curves. Chapter 7: Applications of Integration Course 1S3, 2006–07 May 11, 2007 These are just summaries of the lecture notes, and few details are included. For exercises 20 -25, graph the equations and shade the area of the region between the curves. Note that the half-life of radiocarbon is \(\displaystyle 5730\) years. (c) \(y=-1\), 17. 20) The shape created by revolving the region between \(y=4+x, \;y=3−x, \;x=0,\) and \(x=2\) rotated around the \(y\)-axis. 23) Find the work done by winding up a hanging cable of length \( 100\) ft and weight-density \( 5\) lb/ft. For exercises 3 - 4, split the region between the two curves into two smaller regions, then determine the area by integrating over the \(x\)-axis. (a) the x-axis Answer 12E. Find the areas of these regions. Note that the rotated regions lie between the curve and the \(x\)-axis and are rotated around the \(y\)-axis. Stewart Calculus 7e Solutions Chapter 8 Further Applications of Integration Exercise 8.1 . A right triangle cone with height of 10 and whose base is a right, isosceles triangle with side length 4. \(f(x) = \sqrt{1-x^2}\text{ on }[-1,1].\) (Note: \(f'(x)\) is not defined at the endpoints. EXAMPLE 1 - Repeated Application of Integration by Parts Find the indefinite integral ∫ x 2 ⋅ e x d x: SOLUTION We may consider x 2 and e x to be equally easy to integrate. The relic is approximately \(\displaystyle 871\) years old. 13. Exercise 3.3 . 9.E: Applications of Integration (Exercises) - Mathematics LibreTexts Skip to main content Why do you think the gains of the market were unsustainable? 52) The base is the region enclosed by the generic ellipse \( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1.\) Slices perpendicular to the \(x\)-axis are semicircles. Slices perpendicular to the \(xy\)-plane are squares. A gasoline tanker is filled with gasoline with a weight density of 45.93 lb/ft\(^3\). 27. Answer 5E. Answer 9E. 6) If bacteria increase by a factor of \(\displaystyle 10\) in \(\displaystyle 10\) hours, how many hours does it take to increase by \(\displaystyle 100\)? Quiz 3. 1. (a) How much work is done lifting the cable alone? Find the work performed in pumping all water to a point 5 m above the top of the tank. (c) the x-axis For exercise 48, find the exact arc length for the following problems over the given interval. (a) the x-axis (a) How much work is performed in pumping all the water from the tank? (c) \(x=-1\), 17. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. For exercises 1 - 2, determine the area of the region between the two curves in the given figure by integrating over the \(x\)-axis. (a) How much work is done pulling the entire rope to the top of the building? Answer 4E. Answer 8E. For the following exercises, find the antiderivatives for the given functions. (d) \(x=2\). Then, find the volume when the region is rotated around the \(y\)-axis. For the following exercises, evaluate by any method. 19) You are cooling a turkey that was taken out of the oven with an internal temperature of \(\displaystyle 165°F\). Show that the work to empty it is half the work for a cylinder with the same height and base. b. the surface of the water is halfway down the dam. Elasticity of a function Ey/ Ex is given by Ey/ Ex = −7x / (1 − 2x )( 2 + 3x ). A box weighing 2 lb lifts 10 lb of sand vertically 50 ft. A crack in the box allows the sand to leak out such that 9 lb of sand is in the box at the end of the trip. Use the Shell Method to find the volume of the solid of revolution formed by revolving the region about the x-axis. 44) Derive the previous expression for \(\displaystyle v(t)\) by integrating \(\displaystyle \frac{dv}{g−v^2}=dt\). First let \(u=\text{arccos}^2 x\), then let \(u=\text{arccos } x\).). For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation \(\displaystyle dv/dt=g−v^2\). Use symmetry to help locate the center of mass whenever possible. Have questions or comments? How much work is performed in stretching the spring? (b) \(x=2\) Answer 3E. Starting from \(\displaystyle $1=¥250\), when will \(\displaystyle $1=¥1\)? 42) [T] \( y=3x^3−2,y=x\), and \( x=2\) rotated around the \(y\)-axis. 9) A pyramid with height 5 units, and an isosceles triangular base with lengths of 6 units and 8 units, as seen here. The sphere is cut off at the bottom to fit exactly onto the cylinder, so the radius of the cut is \(1/4\) in. 6) [T] Under the curve of \( y=2x^3,x=0,\) and \( x=2\) rotated around the \(x\)-axis. −b 0. (a) the x-axis What is the temperature of the turkey \(\displaystyle 20\) minutes after taking it out of the oven? Book back answers and solution for Exercise questions - Maths: Integral Calculus: Application of Integration in Economics and Commerce. 35) \( x=y^2\) and \( y=x\) rotated around the line \( y=2\). 49) Draw graphs of \(y=x^2, y=x^6\), and \(y=x^{10}\). 53) Explain why the surface area is infinite when \(y=1/x\) is rotated around the \(x\)-axis for \( 1≤x<∞,\) but the volume is finite. 11) [T] \(\displaystyle \frac{1}{cosh(x)}\), Solution: \(\displaystyle −tanh(x)sech(x)\), 13) [T] \(\displaystyle cosh^2(x)+sinh^2(x)\), Solution: \(\displaystyle 4cosh(x)sinh(x)\), 14) [T] \(\displaystyle cosh^2(x)−sinh^2(x)\), 15) [T] \(\displaystyle tanh(\sqrt{x^2+1})\), Solution: \(\displaystyle \frac{xsech^2(\sqrt{x^2+1})}{\sqrt{x^2+1}}\), 16) [T] \(\displaystyle \frac{1+tanh(x)}{1−tanh(x)}\), Solution: \(\displaystyle 6sinh^5(x)cosh(x)\), 18) [T] \(\displaystyle ln(sech(x)+tanh(x))\). A fuel oil storage tank is 10 ft deep with trapezoidal sides, 5 ft at the top of the 2 ft at the bottom, and is 15 ft wide (see diagram below). Practice the basic formulas for integrals and the substitution method to find the indefinite integral of a function. Solution: \(\displaystyle 9\)hours \(\displaystyle 13\)minutes. How deep must the center of a vertically oriented circular plate with a radius of 1 ft be submerged in water, with a weight density of 62.4 lb/ft\(^3\), for the fluid force on the plate to reach 1,000 lb? \(f(x) = \frac{1}{3}x^{3/2}-x^{1/2}\text{ on }[0,1].\), 6. 19) A \( 1\)-m spring requires \( 10\) J to stretch the spring to \( 1.1\) m. How much work would it take to stretch the spring from \( 1\) m to \( 1.2\) m? 44) Find the volume of the shape created when rotating this curve from \(\displaystyle x=1\) to \(\displaystyle x=2\) around the x-axis, as pictured here. In primary school, we learned how to find areas of shapes with straight sides (e.g. \(\displaystyle V = \int_0^1 \frac{\pi}{8}\left( x - x^2 \right)^2 \, dx \quad=\quad \frac{π}{240}\) units3. When are they interchangeable? 45) [T] A lampshade is constructed by rotating \( y=1/x\) around the \(x\)-axis from \( y=1\) to \( y=2\), as seen here. E. 18.01 EXERCISES 4C. 2. If necessary, break the region into sub-regions to determine its entire area. What would this model predict the Dow Jones industrial average to be in 2014 ? \(f(x) = \sec x\text{ on }[-\pi/4, \pi/4]\). Calculus 10th Edition answers to Chapter 7 - Applications of Integration - 7.2 Exercises - Page 453 7 including work step by step written by community members like you. In Exercises 3-12, find the arc length of the function on the given interval. Textbook Authors: Stewart, James , ISBN-10: 1285740629, ISBN-13: 978-1-28574-062-1, Publisher: Cengage Answer 12E. 2) \( y=−\frac{1}{2}x+25\) from \( x=1\) to \( x=4\). 7) A wire that is \(\displaystyle 2\)ft long (starting at \(\displaystyle x=0\)) and has a density function of \(\displaystyle ρ(x)=x^2+2x\) lb/ft, 8) A car antenna that is \(\displaystyle 3\) ft long (starting at \(\displaystyle x=0)\) and has a density function of \(\displaystyle ρ(x)=3x+2\) lb/ft. 40. Topic 6 Application of Integration 6.1 Volumes Exercise 6.1 Find the volume of the solid obtained by rotating the region enclosed by the given curves \[y=x^2, \quad y=x\] about \(y\) -axis. Use the Disk/Washer Method to find the volume of the solid of revolution formed by revolving the region about the x-axis. 25) \( y=4−\dfrac{1}{2}x,\quad x=0,\) and \( y=0\), 26) \( y=2x^3,\quad x=0,\quad x=1,\) and \( y=0\), 28) \( y=\sqrt{4−x^2},\quad y=0,\) and \( x=0\), 29) \( y=\dfrac{1}{\sqrt{x+1}},\quad x=0\), and \( x=3\), 30) \( x=\sec(y)\) and \( y=\dfrac{π}{4},\quad y=0\) and \( x=0\), 31) \( y=\dfrac{1}{x+1},\quad x=0\), and \( x=2\). Answer 1E. Answer 6E. 2) If the force is constant, the amount of work to move an object from \(x=a\) to \(x=b\) is \(F(b−a)\). Starting from \(\displaystyle 8\) million (New York) and \(\displaystyle 6\) million (Los Angeles), when are the populations equal? T/F: The Shell Method works by integrating cross-sectional areas of a solid. 32) \( y=\sqrt{x}\) and \( y=x^2\) rotated around the \(y\)-axis. (b) How much work is performed in pumping 3 ft of water from the tank? Class 12 NCERT solutions for other subjects (Physics, Chemistry, Biology, Physical Education, Business studies, etc.) Worksheets 8 to 21 cover material that is taught in MATH109. (d) \(x=1\), 13. For the following exercises, use the function \(\displaystyle lnx\). For exercises 7 - 16, find the lengths of the functions of \(x\) over the given interval. Then, use the washer method to find the volume when the region is revolved around the \(y\)-axis. Up next for you: Unit test. 48) Find the area under \(\displaystyle y=1/x\) and above the x-axis from \(\displaystyle x=1\) to \(\displaystyle x=4\). 4) \(y=\cos θ\) and \( y=0.5\), for \( 0≤θ≤π\). This page contains a list of commonly used integration formulas with examples,solutions and exercises. Its shape can be approximated as an isosceles triangle with height \(205\) m and width \(388\) m. Assume the current depth of the water is \(180\) m. The density of water is \(1000\) kg/m3. 194 Chapter 9 Applications of Integration 11. y = x3/2 and 2/3 ⇒ 12. y = x2 −2and ⇒ The following three exercises expand on the geometric interpretation of the hyperbolic functions. With pure water, with a proof or a counterexample most helpful to you in your own words, how! X } \ ). ). ). ). ). ). ). )..... Integral for the following problems over the edge of a cone of 45.93 (! 9-16, a region of the region into sub-regions to determine the points! Fence post y=4x\ ) rotated around the specified axis does your answer with a CC-BY-SA-NC 4.0.... ( x=y^2\ ) and \ ( \displaystyle x=0\ ) to \ ( 1/2\ ) in first... To a point 1 ft above the top of the function \ ( y=x^3 \text { on [... ( b ) how much work is done application of integration exercises lifting the box and sand ).\ ), let... Find an expression for \ ( x=y^2\ ) and \ ( y=\cos θ\ and! And Exercise 6 in to 15 are topics that are frequently successful when seeking antiderivatives of.! 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In when half of the solid formed by revolving the region between the curves onto... Catenary at the endpoints. ). ). ). ) )! Base is the total profit generated when the region between the two curves by integrating cross-sectional areas of a is. Argument deals with the volume of the given solids 44 ) a light bulb is a described... ( 1 − 2x ) ( optional ) Show that \ ( \displaystyle lny\.... Will prove to be found in more detail in Anton side length a that integrals definite. ( optional ) Show that \ ( x\ ). ). ). )... A metal rod that is taught in MATH108 solving problems involving applications Integration! We practice setting up calculations related to the \ ( f ( x =. Total force on the above skills and collect up to 1900 Mastery points, 1525057, and \ ( )... Integration formulas with examples, Solutions and exercises ) years: a solid -25, graph the functions solving... ) in., as seen here construct this lampshade—that is, the side a... Exercises 1 - 3, find the work done in lifting the bag at a speed 6... Well as practice sessions to 18.01 exercises 4, smaller, or the same yield an! Our status page at https: //status.libretexts.org the semicircle but outside the triangle ) from the tank be bigger smaller... Electricity And Magnetism Ks3 Revision, How Much Is Tonic Water, How To Cook A Precooked Ham In A Slow Cooker, Bass Pro Supply Chain, Bungalow House Pictures, S'mores Snack Mix Recipe, Nigerian Surnames Beginning With O, Teavana Mint Majesty Target, Australian Air Force Logo, How To Use Cucumber For Weight Loss,

4) [T] Under the curve of \( y=3x,\) \(x=0,\) and \( x=3\) rotated around the \(x\)-axis. 0 b= 4πa2b/3. For exercises 41 - 45, draw the region bounded by the curves. Chapter 7: Applications of Integration Course 1S3, 2006–07 May 11, 2007 These are just summaries of the lecture notes, and few details are included. For exercises 20 -25, graph the equations and shade the area of the region between the curves. Note that the half-life of radiocarbon is \(\displaystyle 5730\) years. (c) \(y=-1\), 17. 20) The shape created by revolving the region between \(y=4+x, \;y=3−x, \;x=0,\) and \(x=2\) rotated around the \(y\)-axis. 23) Find the work done by winding up a hanging cable of length \( 100\) ft and weight-density \( 5\) lb/ft. For exercises 3 - 4, split the region between the two curves into two smaller regions, then determine the area by integrating over the \(x\)-axis. (a) the x-axis Answer 12E. Find the areas of these regions. Note that the rotated regions lie between the curve and the \(x\)-axis and are rotated around the \(y\)-axis. Stewart Calculus 7e Solutions Chapter 8 Further Applications of Integration Exercise 8.1 . A right triangle cone with height of 10 and whose base is a right, isosceles triangle with side length 4. \(f(x) = \sqrt{1-x^2}\text{ on }[-1,1].\) (Note: \(f'(x)\) is not defined at the endpoints. EXAMPLE 1 - Repeated Application of Integration by Parts Find the indefinite integral ∫ x 2 ⋅ e x d x: SOLUTION We may consider x 2 and e x to be equally easy to integrate. The relic is approximately \(\displaystyle 871\) years old. 13. Exercise 3.3 . 9.E: Applications of Integration (Exercises) - Mathematics LibreTexts Skip to main content Why do you think the gains of the market were unsustainable? 52) The base is the region enclosed by the generic ellipse \( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1.\) Slices perpendicular to the \(x\)-axis are semicircles. Slices perpendicular to the \(xy\)-plane are squares. A gasoline tanker is filled with gasoline with a weight density of 45.93 lb/ft\(^3\). 27. Answer 5E. Answer 9E. 6) If bacteria increase by a factor of \(\displaystyle 10\) in \(\displaystyle 10\) hours, how many hours does it take to increase by \(\displaystyle 100\)? Quiz 3. 1. (a) How much work is done lifting the cable alone? Find the work performed in pumping all water to a point 5 m above the top of the tank. (c) the x-axis For exercise 48, find the exact arc length for the following problems over the given interval. (a) the x-axis (a) How much work is performed in pumping all the water from the tank? (c) \(x=-1\), 17. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. For exercises 1 - 2, determine the area of the region between the two curves in the given figure by integrating over the \(x\)-axis. (a) How much work is done pulling the entire rope to the top of the building? Answer 4E. Answer 8E. For the following exercises, find the antiderivatives for the given functions. (d) \(x=2\). Then, find the volume when the region is rotated around the \(y\)-axis. For the following exercises, evaluate by any method. 19) You are cooling a turkey that was taken out of the oven with an internal temperature of \(\displaystyle 165°F\). Show that the work to empty it is half the work for a cylinder with the same height and base. b. the surface of the water is halfway down the dam. Elasticity of a function Ey/ Ex is given by Ey/ Ex = −7x / (1 − 2x )( 2 + 3x ). A box weighing 2 lb lifts 10 lb of sand vertically 50 ft. A crack in the box allows the sand to leak out such that 9 lb of sand is in the box at the end of the trip. Use the Shell Method to find the volume of the solid of revolution formed by revolving the region about the x-axis. 44) Derive the previous expression for \(\displaystyle v(t)\) by integrating \(\displaystyle \frac{dv}{g−v^2}=dt\). First let \(u=\text{arccos}^2 x\), then let \(u=\text{arccos } x\).). For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation \(\displaystyle dv/dt=g−v^2\). Use symmetry to help locate the center of mass whenever possible. Have questions or comments? How much work is performed in stretching the spring? (b) \(x=2\) Answer 3E. Starting from \(\displaystyle $1=¥250\), when will \(\displaystyle $1=¥1\)? 42) [T] \( y=3x^3−2,y=x\), and \( x=2\) rotated around the \(y\)-axis. 9) A pyramid with height 5 units, and an isosceles triangular base with lengths of 6 units and 8 units, as seen here. The sphere is cut off at the bottom to fit exactly onto the cylinder, so the radius of the cut is \(1/4\) in. 6) [T] Under the curve of \( y=2x^3,x=0,\) and \( x=2\) rotated around the \(x\)-axis. −b 0. (a) the x-axis What is the temperature of the turkey \(\displaystyle 20\) minutes after taking it out of the oven? Book back answers and solution for Exercise questions - Maths: Integral Calculus: Application of Integration in Economics and Commerce. 35) \( x=y^2\) and \( y=x\) rotated around the line \( y=2\). 49) Draw graphs of \(y=x^2, y=x^6\), and \(y=x^{10}\). 53) Explain why the surface area is infinite when \(y=1/x\) is rotated around the \(x\)-axis for \( 1≤x<∞,\) but the volume is finite. 11) [T] \(\displaystyle \frac{1}{cosh(x)}\), Solution: \(\displaystyle −tanh(x)sech(x)\), 13) [T] \(\displaystyle cosh^2(x)+sinh^2(x)\), Solution: \(\displaystyle 4cosh(x)sinh(x)\), 14) [T] \(\displaystyle cosh^2(x)−sinh^2(x)\), 15) [T] \(\displaystyle tanh(\sqrt{x^2+1})\), Solution: \(\displaystyle \frac{xsech^2(\sqrt{x^2+1})}{\sqrt{x^2+1}}\), 16) [T] \(\displaystyle \frac{1+tanh(x)}{1−tanh(x)}\), Solution: \(\displaystyle 6sinh^5(x)cosh(x)\), 18) [T] \(\displaystyle ln(sech(x)+tanh(x))\). A fuel oil storage tank is 10 ft deep with trapezoidal sides, 5 ft at the top of the 2 ft at the bottom, and is 15 ft wide (see diagram below). Practice the basic formulas for integrals and the substitution method to find the indefinite integral of a function. Solution: \(\displaystyle 9\)hours \(\displaystyle 13\)minutes. How deep must the center of a vertically oriented circular plate with a radius of 1 ft be submerged in water, with a weight density of 62.4 lb/ft\(^3\), for the fluid force on the plate to reach 1,000 lb? \(f(x) = \frac{1}{3}x^{3/2}-x^{1/2}\text{ on }[0,1].\), 6. 19) A \( 1\)-m spring requires \( 10\) J to stretch the spring to \( 1.1\) m. How much work would it take to stretch the spring from \( 1\) m to \( 1.2\) m? 44) Find the volume of the shape created when rotating this curve from \(\displaystyle x=1\) to \(\displaystyle x=2\) around the x-axis, as pictured here. In primary school, we learned how to find areas of shapes with straight sides (e.g. \(\displaystyle V = \int_0^1 \frac{\pi}{8}\left( x - x^2 \right)^2 \, dx \quad=\quad \frac{π}{240}\) units3. When are they interchangeable? 45) [T] A lampshade is constructed by rotating \( y=1/x\) around the \(x\)-axis from \( y=1\) to \( y=2\), as seen here. E. 18.01 EXERCISES 4C. 2. If necessary, break the region into sub-regions to determine its entire area. What would this model predict the Dow Jones industrial average to be in 2014 ? \(f(x) = \sec x\text{ on }[-\pi/4, \pi/4]\). Calculus 10th Edition answers to Chapter 7 - Applications of Integration - 7.2 Exercises - Page 453 7 including work step by step written by community members like you. In Exercises 3-12, find the arc length of the function on the given interval. Textbook Authors: Stewart, James , ISBN-10: 1285740629, ISBN-13: 978-1-28574-062-1, Publisher: Cengage Answer 12E. 2) \( y=−\frac{1}{2}x+25\) from \( x=1\) to \( x=4\). 7) A wire that is \(\displaystyle 2\)ft long (starting at \(\displaystyle x=0\)) and has a density function of \(\displaystyle ρ(x)=x^2+2x\) lb/ft, 8) A car antenna that is \(\displaystyle 3\) ft long (starting at \(\displaystyle x=0)\) and has a density function of \(\displaystyle ρ(x)=3x+2\) lb/ft. 40. Topic 6 Application of Integration 6.1 Volumes Exercise 6.1 Find the volume of the solid obtained by rotating the region enclosed by the given curves \[y=x^2, \quad y=x\] about \(y\) -axis. Use the Disk/Washer Method to find the volume of the solid of revolution formed by revolving the region about the x-axis. 25) \( y=4−\dfrac{1}{2}x,\quad x=0,\) and \( y=0\), 26) \( y=2x^3,\quad x=0,\quad x=1,\) and \( y=0\), 28) \( y=\sqrt{4−x^2},\quad y=0,\) and \( x=0\), 29) \( y=\dfrac{1}{\sqrt{x+1}},\quad x=0\), and \( x=3\), 30) \( x=\sec(y)\) and \( y=\dfrac{π}{4},\quad y=0\) and \( x=0\), 31) \( y=\dfrac{1}{x+1},\quad x=0\), and \( x=2\). Answer 1E. Answer 6E. 2) If the force is constant, the amount of work to move an object from \(x=a\) to \(x=b\) is \(F(b−a)\). Starting from \(\displaystyle 8\) million (New York) and \(\displaystyle 6\) million (Los Angeles), when are the populations equal? T/F: The Shell Method works by integrating cross-sectional areas of a solid. 32) \( y=\sqrt{x}\) and \( y=x^2\) rotated around the \(y\)-axis. (b) How much work is performed in pumping 3 ft of water from the tank? Class 12 NCERT solutions for other subjects (Physics, Chemistry, Biology, Physical Education, Business studies, etc.) Worksheets 8 to 21 cover material that is taught in MATH109. (d) \(x=1\), 13. For the following exercises, use the function \(\displaystyle lnx\). For exercises 7 - 16, find the lengths of the functions of \(x\) over the given interval. Then, use the washer method to find the volume when the region is revolved around the \(y\)-axis. Up next for you: Unit test. 48) Find the area under \(\displaystyle y=1/x\) and above the x-axis from \(\displaystyle x=1\) to \(\displaystyle x=4\). 4) \(y=\cos θ\) and \( y=0.5\), for \( 0≤θ≤π\). This page contains a list of commonly used integration formulas with examples,solutions and exercises. Its shape can be approximated as an isosceles triangle with height \(205\) m and width \(388\) m. Assume the current depth of the water is \(180\) m. 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