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6.1E: Exercises for Section 6.1

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For exercises 1 - 2, determine the area of the region between the two curves in the given figure by integrating over the x-axis.

1) y=x23 and y=1

CNX_Calc_Figure_06_01_201.jpeg

Answer:
323units2

2) y=x2 and y=3x+4

CNX_Calc_Figure_06_01_202.jpeg

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. Note that you will have two integrals to solve.

3) y=x3 and y=x2+x

CNX_Calc_Figure_06_01_203.jpeg

Answer:
1312units2

4) y=cosθ and y=0.5, for 0θπ

CNX_Calc_Figure_06_01_204.jpeg

 

For exercises 5-6, determine the area of the region between the two curves by integrating over the y-axis.

5) x=y2 and x=9

CNX_Calc_Figure_06_01_205.jpeg

Answer:
36units2

6) y=x and x=y2

CNX_Calc_Figure_06_01_206.jpeg

For exercises 7 - 13, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis.

7) y=x2 and y=x2+18x

Answer:

CNX_Calc_Figure_06_01_207.jpeg

243 square units

8) y=1x,y=1x2, and x=3

9) y=cosx and y=cos2x on x[π,π]

Answer:

CNX_Calc_Figure_06_01_209.jpeg

4 square units

10) y=ex,y=e2x1, and x=0

11) y=ex,y=ex,x=1 and x=1

Answer:

CNX_Calc_Figure_06_01_211.jpeg

2(e1)2eunits2

12) y=e,y=ex, and y=ex

13) y=|x| and y=x2

Answer:

CNX_Calc_Figure_06_01_213.jpeg

13units2

For exercises 14 - 19, graph the equations and shade the area of the region between the curves. If necessary, break the region into sub-regions to determine its entire area.

14) y=sin(πx),y=2x, and x>0

15) y=12x,y=x, and y=1

Answer:

CNX_Calc_Figure_06_01_215.jpeg

343units2

16) y=sinx and y=cosx over x[π,π]

17) y=x3 and y=x22x over x[1,1]

Answer:

CNX_Calc_Figure_06_01_217.jpeg

52units2

18) y=x2+9 and y=10+2x over x[1,3]

19) y=x3+3x and y=4x

Answer:

CNX_Calc_Figure_06_01_219.jpeg

12units2

For exercises 20 -25, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the y-axis.

20) x=y3 and x=3y2

21) x=y and x=y3y

Answer:

CNX_Calc_Figure_06_01_221.jpeg

92units2

22) x=3+y2 and x=yy2

23) y2=x and x=y+2

Answer:

CNX_Calc_Figure_06_01_223.jpeg

92units2

24) x=|y| and 2x=y2+2

25) x=siny,x=cos(2y),y=π/2, and y=π/2

Answer:

CNX_Calc_Figure_06_01_225.jpeg

332units2

For exercises 26 - 37, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the x-axis or y-axis, whichever seems more convenient.

26) x=y4 and x=y5

27) y=xex,y=ex,x=0, and x=1.

Answer:

CNX_Calc_Figure_06_01_227.jpeg

(e2)units2

28) y=x6 and y=x4

29) x=y3+2y2+1 and x=y2+1

Answer:

CNX_Calc_Figure_06_01_229.jpeg

274units2

30) y=|x| and y=x21

31) y=43x and y=1x

Answer:

CNX_Calc_Figure_06_01_231.jpeg

(43ln(3))units2

32) y=sinx,x=π/6,x=π/6, and y=cos3x

33) y=x23x+2 and y=x32x2x+2

Answer:
CNX_Calc_Figure_06_01_233.jpg
12
square units

34) y=2cos3(3x),y=1,x=π4, and x=π4

35) y+y3=x and 2y=x

Answer:

CNX_Calc_Figure_06_01_235.jpeg

12 square units

36) y=1x2 and y=x21

37) y=arccosx,y=arcsinx,x=1, and x=1

Answer:

CNX_Calc_Figure_06_01_237.jpeg

2(2π) square units

 

For exercises 38 - 47, find the exact area of the region bounded by the given equations if possible. 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.

38) [T] x=ey and y=x2

39) [T] y=x2 and y=1x2

Answer:
1.067 square units

40) [T] y=3x2+8x+9 and 3y=x+24

41) [T] x=4y2 and y2=1+x2

Answer:
0.852 square units

42) [T] x2=y3 and x=3y

43) [T] y=sin3x+2,y=tanx,x=1.5, and x=1.5

Answer:
7.523 square units

44) [T] y=1x2 and y2=x2

45) [T] y=1x2 and y=x2+2x+1

Answer:
3π412 square units

46) [T] x=4y2 and x=1+3y+y2

47) [T] y=cosx,y=ex,x=π, andx=0

Answer:
1.429 square units

48) The largest triangle with a base on the x-axis that fits inside the upper half of the unit circle y2+x2=1 is given by y=1+x and y=1x. See the following figure. What is the area inside the semicircle but outside the triangle?

CNX_Calc_Figure_06_01_238.jpeg

49) A factory selling cell phones has a marginal cost function C(x)=0.01x23x+229, where x represents the number of cell phones, and a marginal revenue function given by R(x)=4292x. Find the area between the graphs of these curves and x=0. What does this area represent?

Answer:
$33,333.33 total profit for 200 cell phones sold

50) An amusement park has a marginal cost function C(x)=1000ex+5, where x represents the number of tickets sold, and a marginal revenue function given by R(x)=600.1x. Find the total profit generated when selling 550 tickets. Use a calculator to determine intersection points, if necessary, to two decimal places.

51) The tortoise versus the hare: The speed of the hare is given by the sinusoidal function H(t)=1cos((πt)/2) whereas the speed of the tortoise is T(t)=(1/2)arctan(t/4), where t is time measured in hours and the speed is measured in miles per hour. 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. What does it represent? Use a calculator to determine the intersection points, if necessary, accurate to three decimal places.

Answer:
3.263 mi represents how far ahead the hare is from the tortoise

52) The tortoise versus the hare: The speed of the hare is given by the sinusoidal function H(t)=(1/2)(1/2)cos(2πt) whereas the speed of the tortoise is T(t)=t, where t is time measured in hours and speed is measured in kilometers per hour. If the race is over in 1 hour, who won the race and by how much? Use a calculator to determine the intersection points, if necessary, accurate to three decimal places.

For exercises 53 - 55, find the area between the curves by integrating with respect to x and then with respect to y. Is one method easier than the other? Do you obtain the same answer?

53) y=x2+2x+1 and y=x23x+4

Answer:
34324 square units

54) y=x4 and x=y5

55) x=y22 and x=2y

Answer:
43 square units

For exercises 56 - 57, solve using calculus, then check your answer with geometry.

56) Determine the equations for the sides of the square that touches the unit circle on all four sides, as seen in the following figure. Find the area between the perimeter of this square and the unit circle. Is there another way to solve this without using calculus?

CNX_Calc_Figure_06_01_239.jpeg

57) Find the area between the perimeter of the unit circle and the triangle created from y=2x+1,y=12x and y=35, as seen in the following figure. Is there a way to solve this without using calculus?

CNX_Calc_Figure_06_01_240.jpeg

Answer:
(π3225) square units

Contributors

  • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

  • Paul Seeburger (Monroe Community College) edited to use alternate notation for inverse trig functions.

6.1E: Exercises for Section 6.1 is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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