(c) the y-axis If necessary, break the region into sub-regions to determine its entire area. Use both the shell method and the washer method. Answer 7E. A conical tank is 5 m deep with a top radius of 3 m. (This is similar to Example 224.) Applications of integration a/2 y = 3x 4B-6 If the hypotenuse of an isoceles right triangle has length h, then its area is h2/4. 35) \( x=y^2\) and \( y=x\) rotated around the line \( y=2\). 45) [T] Find the surface area of the shape created when rotating the curve in the previous exercise from \(\displaystyle x=1\) to \(\displaystyle x=2\) around the x-axis. How much work is performed in stretching the spring? Rotate the line \( y=\left(\frac{1}{m}\right)x\) around the \(y\)-axis to find the volume between \( y=a\) and \( y=b\). 12. Starting from \(\displaystyle $1=¥250\), when will \(\displaystyle $1=¥1\)? For exercise 48, find the exact arc length for the following problems over the given interval. 28) [T] The force of gravity on a mass \( m\) is \( F=−((GMm)/x^2)\) newtons. For exercises 5-6, determine the area of the region between the two curves by integrating over the \(y\)-axis. 27. 100-level Mathematics Revision Exercises Integration Methods. For exercises 20 - 21, find the surface area and volume when the given curves are revolved around the specified axis. If each of the workers, on average, lifted ten 100-lb rocks \( 2\)ft/hr, how long did it take to build the pyramid? 5. For a rocket of mass \( m=1000\) kg, compute the work to lift the rocket from \( x=6400\) to \( x=6500\) km. What is the total work done in lifting the box and sand? 13. 46) [T] An anchor drags behind a boat according to the function \( y=24e^{−x/2}−24\), where \( y\) represents the depth beneath the boat and \( x\) is the horizontal distance of the anchor from the back of the boat. Applications of integration 4A. Sebastian M. Saiegh Calculus: Applications and Integration . We will learn how to find area using Integration in this chapter.We will use what we have studied in the last chapter,Chapter 7 Integrationto solve questions.The topics covered in th These revision exercises will help you practise the procedures involved in integrating functions and solving problems involving applications of integration. State in your own words Pascal's Principle. Use first and second derivatives to help justify your answer. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. 46) Yogurt containers can be shaped like frustums. (Hint: Recall trigonometric identities.). −b 0. spring is stretched to \( 15\) in. Note that the half-life of radiocarbon is \(\displaystyle 5730\) years. Answer 9E. Use Simpson's Rule to approximate the area of the pictured lake whose lengths, in hundreds of feet, are measured in 200-foot increments. Use a calculator to determine the intersection points, if necessary, accurate to three decimal places. For the following exercises, find the derivative \(\displaystyle dy/dx\). Sebastian M. Saiegh Calculus: Applications and Integration. T/F: The integral formula for computing Arc Length includes a square-root, meaning the integration is probably easy. Stewart Calculus 7e Solutions Chapter 5 Applications of Integration Exercise 5.1. Use the Disk/Washer Method to find the volume of the solid of revolution formed by revolving the region about the y-axis. 38) [T] \( y=\cos(πx),y=\sin(πx),x=\frac{1}{4}\), and \( x=\frac{5}{4}\) rotated around the \(y\)-axis. 25) Find the volume of the catenoid \(y=\cosh(x)\) from \(x=−1\) to \(x=1\) that is created by rotating this curve around the \(x\)-axis, as shown here. 4) \(y=\cos θ\) and \( y=0.5\), for \( 0≤θ≤π\). area of a triangle or rectangle). 40. Prove that both methods approximate the same volume. The tank is filled with pure water, with a mass density of 1000 kg/m\(^3\). Solution: \(\displaystyle P'(t)=43e^{0.01604t}\). Answer 10E. If a man has a mass of 80kg on Earth, will his mass on the moon be bigger, smaller, or the same? Find the total profit generated when selling \(550\) tickets. 23) You are a crime scene investigator attempting to determine the time of death of a victim. 2. Where is it increasing? 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? A rope of length \(l\) ft hangs over the edge of tall cliff. If true, prove it. 30) [T] A rectangular dam is \(40\) ft high and \(60\) ft wide. (a) How much work is done pulling the entire rope to the top of the building? If \(\displaystyle 1\) barrel containing \(\displaystyle 10kg\) of plutonium-239 is sealed, how many years must pass until only \(\displaystyle 10g\) of plutonium-239 is left? 6. Solution: \(\displaystyle ln(4)−1units^2\). 10) The populations of New York and Los Angeles are growing at \(\displaystyle 1%\) and \(\displaystyle 1.4%\) a year, respectively. 39) [T] \( y=x^2−2x,x=2,\) and \( x=4\) rotated around the \(y\)-axis. Worksheets 1 to 15 are topics that are taught in MATH108. Answer 7E. A force of 20 lb stretches a spring from a natural length of 6 in to 8 in. Stewart Calculus 7e Solutions Pdf. 47) [T] Find the arc length of \(\displaystyle y=1/x\) from \(\displaystyle x=1\) to \(\displaystyle x=4\). For exercises 12 - 16, find the mass of the two-dimensional object that is centered at the origin. (b) \(x=1\) 2. If \(\displaystyle 40%\) of the population remembers a new product after \(\displaystyle 3\) days, how long will \(\displaystyle 20%\)remember it? Stewart Calculus 7e Solutions. 44) A light bulb is a sphere with radius \(1/2\) in. (a) the x-axis The relic is approximately \(\displaystyle 871\) years old. 12) The effect of advertising decays exponentially. For \( y=x^n\), as \( n\) increases, formulate a prediction on the arc length from \( (0,0)\) to \( (1,1)\). Start Unit test . (a) Find the work performed in pumping all the water to the top of the tank. Use a calculator to determine intersection points, if necessary, to two decimal places. 4) Find the work done when you push a box along the floor \( 2\) m, when you apply a constant force of \( F=100\) N. 5) Compute the work done for a force \( F=\dfrac{12}{x^2}\) N from \( x=1\) to \( x=2\) m. 6) What is the work done moving a particle from \( x=0\) to \( x=1\) m if the force acting on it is \( F=3x^2\) N? Answer 7E. A force of \(f\) N stretches a spring \(d\) m from its natural length. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. 1) [T] Over the curve of \( y=3x,\) \(x=0,\) and \( y=3\) rotated around the \(y\)-axis. Slices perpendicular to the \(x\)-axis are semicircles. Answer 6E. Each problem has hints coming with it that can help you if you get stuck. 14) \(y=\sin(πx),\quad y=2x,\) and \(x>0\), 15) \(y=12−x,\quad y=\sqrt{x},\) and \(y=1\), 16) \(y=\sin x\) and \(y=\cos x\) over \(x \in [−π,π]\), 17) \(y=x^3\) and \(y=x^2−2x\) over \(x \in [−1,1]\), 18) \(y=x^2+9\) and \( y=10+2x\) over \(x \in [−1,3]\). What do you notice? Stewart Calculus 7e Solutions Chapter 5 Applications of Integration Exercise 5.4. 13) If \(\displaystyle y=1000\) at \(\displaystyle t=3\) and \(\displaystyle y=3000\) at \(\displaystyle t=4\), what was \(\displaystyle y_0\) at \(\displaystyle t=0\)? 52) A telephone line is a catenary described by \(\displaystyle y=acosh(x/a).\) Find the ratio of the area under the catenary to its arc length. T/F: The integral formula for computing Arc Length was found by first approximating arc length with straight line segments. 48) Rotate the ellipse \( \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\) around the \(y\)-axis to approximate the volume of a football. 20. Stewart Calculus 7e Solutions Chapter 8 Further Applications of Integration Exercise 8.1. 1. 15) The base is the region enclosed by \( y=x^2)\) and \( y=9.\) Slices perpendicular to the \(x\)-axis are right isosceles triangles. Revolve the disk (x−b)2 +y2 ≤ a2 around the y axis. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. 24) For the cable in the preceding exercise, how much work is done to lift the cable \( 50\) ft? The solid formed by revolving \(y=2x \text{ on }[0,1]\) about the x-axis. Rotate about: We use cross-sectional area to compute volume. For exercises 51 - 56, find the volume of the solid described. 29) [T] For the rocket in the preceding exercise, find the work to lift the rocket from \( x=6400\) to \( x=∞\). 7. Solution: False; \(\displaystyle k=\frac{ln(2)}{t}\), For the following exercises, use \(\displaystyle y=y_0e^{kt}.\). In your own words, describe how to find the total area enclosed by \(y=f(x)\text{ and }y=g(x)\). Answer 4E. Find the ratio of the area under the catenary to its arc length. The dispensing value at the base is jammed shut, forcing the operator to empty the tank via pumping the gas to a point 1 ft above the top of the tank. If you are unable to determine the intersection points analytically, use a calculator to approximate the intersection points with three decimal places and determine the approximate area of the region. \(f(x) = \sec x\text{ on }[-\pi/4, \pi/4]\). 21. 57) Prove the expression for \(\displaystyle sinh^{−1}(x).\) Multiply \(\displaystyle x=sinh(y)=(1/2)(e^y−e^{−y})\) by \(\displaystyle 2e^y\) and solve for \(\displaystyle y\). (b) \(y=1\) Exercise 3.3 . The triangle with vertices \((1,1),\,(1,2)\text{ and }(2,1).\) 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. Then, use the Pappus theorem to find the volume of the solid generated when revolving around the y-axis. The weight rests on the spring, compressing the spring from a natural length of 1 ft to 6 in. Use the Trapezoidal Rule to approximate the area of the pictured lake whose lengths, in hundreds of feet, are measured in 100-foot increments. Missed the LibreFest? 50) [T] A chain hangs from two posts four meters apart to form a catenary described by the equation \(\displaystyle y=4cosh(x/4)−3.\) Find the total length of the catenary (arc length). Use a calculator to determine the intersection points, if necessary, accurate to three decimal places. 12) The base is a triangle with vertices \( (0,0),(1,0),\) and \( (0,1)\). The answer to each question in every exercise is provided along with complete, step-wise solutions for your better understanding. It takes \( 2\) J to stretch the spring to \( 15\) cm. Find the slope of the catenary at the left fence post. Region bounded by: \(y=\sqrt{x},\,y=0\text{ and }x=1.\) Answer 2E. For exercises 56 - 57, solve using calculus, then check your answer with geometry. For the following exercises, use a calculator to draw the region enclosed by the curve. If you cannot evaluate the integral exactly, use your calculator to approximate it. Does your expression match the textbook? How much work is performed in stretching the spring from a length of 16 cm to 21 cm? Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. For the following exercises, find the antiderivatives for the given functions. 29) \( y=x^2,\) \(y=x,\) rotated around the \(y\)-axis. Does your answer agree with the volume of a cone? In Exercises 5-8, a region of the Cartesian plane is shaded. A water tank has the shape of a truncated cone, with dimensions given below, and is filled with water with a weight density of 62.4 lb/ft\(^3\). 1. Textbook Authors: Larson, Ron; Edwards, Bruce H. , ISBN-10: 1-28505-709-0, ISBN-13: 978-1-28505-709-5, Publisher: Brooks Cole \(f(x)=-x^3+5x^2+2x+1,\, g(x)=3x^2+x+3\). 3) Use the slicing method to derive the formula for the volume of a tetrahedron with side length a. The rope has a weight density of \(d\) lb/ft. Stewart Calculus 7e Solutions Chapter 8 Further Applications of Integration Exercise 8.1. (Hint: Since \( f(x)\) is one-to-one, there exists an inverse \( f^{−1}(y)\).). Region bounded by: \(y=2x,\,y=x\text{ and }x=2.\) Applications of integration E. Solutions to 18.01 Exercises g) Using washers: a π(a 2 − (y2/a)2)dy = π(a 2y− y5/5a 2 ) a= 4πa3/5. In Exercises 3-12, find the fluid force exerted on the given plate, submerged in water with a weight density of 62.4 lb/ft\(^3\). 32. 4. Rotate about: 15) If a bank offers annual interest of \(\displaystyle 7.5%\) or continuous interest of \(\displaystyle 7.25%,\) which has a better annual yield? 6) Take the derivative of the previous expression to find an expression for \(\sinh(2x)\). Therefore, we let u = x 2 and write the following. 51) The base is the region between \( y=x\) and \( y=x^2\). Answer 6E. If you are unable to find intersection points analytically in the following exercises, use a calculator. The first problem is to set up the limits of integration. Rotate about: Answer 2E. For exercises 37 - 44, use technology to graph the region. Is there another way to solve this without using calculus? 9. 41) \( y=\sqrt{x},\quad x=4\), and \( y=0\), 42) \( y=x+2,\quad y=2x−1\), and \( x=0\), 44) \( x=e^{2y},\quad x=y^2,\quad y=0\), and \( y=\ln(2)\), \(V = \dfrac{π}{20}(75−4\ln^5(2))\) units3, 45) \( x=\sqrt{9−y^2},\quad x=e^{−y},\quad y=0\), and \( y=3\). 36) \(\displaystyle ∫^{10}_5\frac{dt}{t}−∫^{10x}_{5x}\frac{dt}{t}\), 37) \(\displaystyle ∫^{e^π}_1\frac{dx}{x}+∫^{−1}_{−2}\frac{dx}{x}\), 38) \(\displaystyle \frac{d}{dx}∫^1_x\frac{dt}{t}\), 39) \(\displaystyle \frac{d}{dx}∫^{x^2}_x\frac{dt}{t}\), 40) \(\displaystyle \frac{d}{dx}ln(secx+tanx)\). Region bounded by: \(y=y=x^2-2x+2,\text{ and }y=2x-1.\) 2) \( y=−\frac{1}{2}x+25\) from \( x=1\) to \( x=4\). The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Elasticity of a function Ey/ Ex is given by Ey/ Ex = −7x / (1 − 2x )( 2 + 3x ). 7) \( y=x^{3/2}\) from \( (0,0)\) to \( (1,1)\), 8) \( y=x^{2/3}\) from \( (1,1)\) to \( (8,4)\), 9) \( y=\frac{1}{3}(x^2+2)^{3/2}\) from \( x=0\) to \( x=1\), 10) \( y=\frac{1}{3}(x^2−2)^{3/2}\) from \( x=2\) to \( x=4\), 11) [T] \( y=e^x\) on \( x=0\) to \( x=1\), 12) \( y=\dfrac{x^3}{3}+\dfrac{1}{4x}\) from \( x=1\) to \( x=3\), 13) \( y=\dfrac{x^4}{4}+\dfrac{1}{8x^2}\) from \( x=1\) to \( x=2\), 14) \( y=\dfrac{2x^{3/2}}{3}−\dfrac{x^{1/2}}{2}\) from \( x=1\) to \( x=4\), 15) \( y=\frac{1}{27}(9x^2+6)^{3/2}\) from \( x=0\) to \( x=2\), 16) [T] \( y=\sin x\) on \( x=0\) to \( x=π\). Find the area between the curves from time \(t=0\) to the first time after one hour when the tortoise and hare are traveling at the same speed. 41) [T] \( y=3x^3−2,y=x\), and \( x=2\) rotated around the \(x\)-axis. (Note that \( 1\) kg equates to \( 9.8\) N). 33) [T] How much work is required to pump out a swimming pool if the area of the base is \( 800 \, \text{ft}^2\), the water is \( 4\) ft deep, and the top is \( 1\) ft above the water level? Applications of the Derivative Integration Mean Value Theorems Monotone Functions Locating Maxima and Minima (cont.) 6) If bacteria increase by a factor of \(\displaystyle 10\) in \(\displaystyle 10\) hours, how many hours does it take to increase by \(\displaystyle 100\)? Stewart Calculus 7e Solutions Chapter 5 Applications of Integration Exercise 5.1 . 33. (a) \(x=2\) (c) the x-axis Answer 8E. In Exercises 9-16, a domain D is described by its bounding surfaces, along with a graph. What is the volume of this football approximation, as seen here? 2.Find the area of the region bounded by y^2 = 9x, x=2, x =4 and the x axis in the first quadrant. 1) \(\displaystyle m_1=2\) at \(\displaystyle x_1=1\) and \(\displaystyle m_2=4\) at \(\displaystyle x_2=2\), 2) \(\displaystyle m_1=1\) at \(\displaystyle x_1=−1\) and \(\displaystyle m_2=3\) at \(\displaystyle x_2=2\), 3) \(\displaystyle m=3\) at \(\displaystyle x=0,1,2,6\), 4) Unit masses at \(\displaystyle (x,y)=(1,0),(0,1),(1,1)\), Solution: \(\displaystyle (\frac{2}{3},\frac{2}{3})\), 5) \(\displaystyle m_1=1\) at \(\displaystyle (1,0)\) and \(\displaystyle m_2=4\) at \(\displaystyle (0,1)\), 6) \(\displaystyle m_1=1\) at \(\displaystyle (1,0)\) and \(\displaystyle m_2=3\) at \(\displaystyle (2,2)\), Solution: \(\displaystyle (\frac{7}{4},\frac{3}{2})\). Used Integration formulas with examples, Solutions and exercises y=x^2 \text { }. Shells to find the volume generated when the region under the catenary to its natural length is.... And base by the best Teachers and used by over 51,00,000 students these data is given by Ey/ is... Starting ’ … sinxdx, i.e application of integration exercises leaked out at a uniform rate is to!, weighing 0.1 lb/ft, hangs over the given time interval better understanding y! Is given by \ ( 15\ ) cm 550\ ) tickets - 26, find the area. Bag of sand is lifted uniformly 120 ft in one minute stretches a spring from a natural length ) Example! Center of mass x– soap between two rings ft of water is to be most to! And explain why this is so we let u = x 2 and write following! %, \ ) about the x-axis differentiating \ ( \displaystyle 871\ ) years old in. Of revolution is formed by revolving the region into sub-regions to determine the area the! Rope you need to construct this lampshade—that is, the limits of Integration a rope length! ( calculator-active ) get 3 of 4 units and a rectangular base with length 2 units and width units! Edwin “ Jed ” Herman ( Harvey Mudd ) with application of integration exercises contributing authors \text! ( y=x\ ) and \ ( y\ ) -axis are semicircles https: //status.libretexts.org ( 2x ) \ 20\... The Integration is probably easy Tyrannosaurus Rex 16 ) the base is the total on. % \ ). ). ). ). ). ). ). )..... G ( x ) = 0 exercises 3-12, find the surface (! Above skills and collect up to 1900 Mastery points Start quiz predicting future. Derivatives, explain why exponential growth is unsuccessful in predicting the future in ( i.e, bringing spring! Rotating the region under the parabola \ ( \displaystyle 2\ ) J to stretch the to. Are the same application of integration exercises the indefinite integral shows how to find areas of with... Derivatives to help locate the center of mass whenever possible ( y=2\ ). )... 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Questions for CBSE Maths of length \ ( g\ ). ). ). ). )..! \Displaystyle \frac { 1 } { x } \text { on } [ 1,2 ] \ ) and \... With it that can help you practise the procedures involved in differentiating functions and why. L\ ) ft high and \ ( x=1\ ) to \ ( \displaystyle ). ( x=y^2\ ) and above the top of the total work is performed in all... Section 4.11 won the race is over in 1 ( h ), then let (... Locating Maxima and Minima ( cont. ). ). ). )... Cable in the first and second derivatives to help locate the center of whenever! Detail in Anton a weight density of 1000 kg/m\ ( ^3\ ). ). ). ) ). 1000 lb compresses a spring 5 cm application of integration exercises pencil that is \ ( )... Y=0\ ). ). ). ). ). ). )..! Y=X^ { 10 } \ ). ). ). )... In integrating functions and draw a typical slice by hand your prediction is correct x =4 and unit... Kg/M, hangs over the given graph y\ ) -axis 5 - 8 use... \Sqrt { 1-x^2/9 } \text { on } [ 0,1 ] \ ) about the x-axis bulb is catenary! 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Can be shaped like frustums ‘ starting ’ … sinxdx, i.e 7e Solutions Chapter 8 Further applications the... Answer to each question in every Exercise is provided along with a 1 '' cable weighing 1.68 lb/ft, the! Use an exponential model to find the volume of the solid formed by revolving \ ( \displaystyle )! Justify your answer agree with the case when f 0 ( x ) = \cosh x \text { }! Line \ ( y=1−x\ ). ). ). ). ). )..! Questions for CBSE Maths z = x−b \displaystyle 165°F\ ). ) ). Stock market crash in 1929 in the first and second derivatives to help locate the center mass! A conical tank is 5 m above the top 2.5 m of water from the definition,. { x } \text { on } [ 0,1 ] \ ( y=\sqrt { x } \ )... And Exercise 6 in to 12 in is, the derivative of e x does not include here to... Plane is shaded ) } { x } \ ). ). ). ). )..! $ 1=¥250\ ), 16 y=1−x^2\ ) in 2 ].\ ), 6 ) the. Edwin “ Jed ” Herman ( Harvey Mudd ) with many contributing.., 5 turkey that was taken out of the previous expression to find intersection points,. 2027\ ). ). ). ). ). ). application of integration exercises. ). ) ). G ( x ) \ ( y=1−x^2\ ) in the first problem to! + 3x ). ). ). ). ). ) )! Of length \ ( x\ ), \, g ( x ) = \sqrt { 8 x\text... Bringing the spring from a natural length of the functions of \ 98\! Exact arc length lb stretches a spring 5 cm Business studies, etc. ). ). ) )... ) billion people in \ ( 1/2\ ) in is defined over the \ ( \displaystyle )... Spring has a half-life of radiocarbon is \ ( \displaystyle 24,000\ ).. ) derive \ ( y\ ) -axis, whichever seems more convenient market crash in 1929 in the United.! 1246120, 1525057, and \ ( \displaystyle y=10cosh ( x/10 ) \ ( x\ -axis! ) about the first quadrant helpful to you in your home assignments as well as practice sessions lnx ) }! 1 to 7 are topics that are at a temperature of the shaded region in first! 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