5.8.1.1: Exercises for Section 6.1

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Anup Lamichhane
Ohio Northern University

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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) $y=x^2−3$ and $y=1$

$This figure is has two graphs. They are the functions f(x) = x^2-3and g(x)=1. In between these graphs is a shaded region, bounded above by g(x) and below by f(x). The shaded area is between x=-2 and x=2.$

Answer: $\dfrac{32}{3} \, \text{units}^2$

2) $y=x^2$ and $y=3x+4$

$This figure is has two graphs. They are the functions f(x) = x^2 and g(x)= 3x+4. In between these graphs is a shaded region, bounded above by g(x) and below by g(x).$

For exercise 3, 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) $y=x^3$ and $ y=x^2+x$

$This figure is has two graphs. They are the functions f(x) = x^3 and g(x)= x^2+x. These graphs intersect twice. The regions between the intersections are shaded. The first region is bounded above by f(x) and below by g(x). The second region is bounded above by g(x) and below by f(x).$

Answer: $\dfrac{13}{12}\, \text{units}^2$

For exercises 4-5, determine the area of the region between the two curves by integrating over the $y$-axis.

4) $x=y^2$ and $x=9$

$This figure is has two graphs. They are the equations x=y^2 and x=9. The region between the graphs is shaded. It is horizontal, between the y-axis and the line x=9.$

Answer: $36 \, \text{units}^2$

5) $y=x$ and $ x=y^2$

$This figure is has two graphs. They are the equations y=x and x=y^2. The region between the graphs is shaded, bounded above by x=y^2 and below by y=x.$

For exercises 6 - 11, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $x$-axis.

6) $y=x^2$ and $y=−x^2+18x$

Answer

$This figure is has two graphs. They are the functions f(x)=x^2 and g(x)=-x^2+18x. The region between the graphs is shaded, bounded above by g(x) and below by f(x). It is in the first quadrant.$

243 square units

7) $y=\dfrac{1}{x}, \quad y=\dfrac{1}{x^2}$, and $x=3$

8) $y=e^x,\quad y=e^{2x−1}$, and $x=0$

9) $y=e^x, \quad y=e^{−x}, \quad x=−1$ and $x=1$

Answer

$This figure is has two graphs. They are the functions f(x)=e^x and g(x)=e^-x. There are two shaded regions. In the second quadrant the region is bounded by x=-1, g(x) above and f(x) below. The second region is in the first quadrant and is bounded by f(x) above, g(x) below, and x=1.$

$\dfrac{2(e−1)^2}{e}\, \text{units}^2$

10) $ y=e, \quad y=e^x,$ and $y=e^{−x}$

11) $y=|x|$ and $y=x^2$

Answer

$This figure is has two graphs. They are the functions f(x)=x^2 and g(x)=absolute value of x. There are two shaded regions. The first region is in the second quadrant and is between g(x) above and f(x) below. The second region is in the first quadrant and is bounded above by g(x) and below by f(x).$

$\dfrac{1}{3}\, \text{units}^2$

For exercises 12 - 15, 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.

12) $y=12−x,\quad y=\sqrt{x},$ and $y=1$

Answer

$This figure is has three graphs. They are the functions f(x)=squareroot of x, y=12-x, and y=1. The region between the graphs is shaded, bounded above and to the left by f(x), above and to the right by the line y=12-x, and below by the line y=1. It is in the first quadrant.$

$\dfrac{34}{3}\, \text{units}^2$

13) $y=x^3$ and $y=x^2−2x$ over $x \in [−1,1]$

Answer

$This figure is has two graphs. They are the functions f(x)=x^3 and g(x)=x^2-2x. There are two shaded regions between the graphs. The first region is bounded to the left by the line x=-2, above by g(x) and below by f(x). The second region is bounded above by f(x), below by g(x) and to the right by the line x=2.$

$\dfrac{5}{2}\, \text{units}^2$

14) $y=x^2+9$ and $ y=10+2x$ over $x \in [−1,3]$

15) $y=x^3+3x$ and $y=4x$

Answer

$This figure is has two graphs. They are the functions f(x)=x^3+3x and g(x)=4x. There are two shaded regions between the graphs. The first region is bounded above by f(x) and below by g(x). The second region is bounded above by g(x), below by f(x).$

$\dfrac{1}{2}\, \text{units}^2$

For exercises 16 - 20, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the $y$-axis.

16) $x=y^3$ and $ x = 3y−2$

17) $x=2y$ and $ x=y^3−y$

Answer

$This figure is has two graphs. They are the equations x=2y and x=y^3-y. The graphs intersect in the third quadrant and again in the first quadrant forming two closed regions in between them.$

$\dfrac{9}{2}\, \text{units}^2$

18) $x=−3+y^2$ and $ x=y−y^2$

19) $y^2=x$ and $x=y+2$

Answer

$This figure is has two graphs. They are the equations x=y+2 and y^2=x. The graphs intersect, forming a region in between them$

$\dfrac{9}{2}\, \text{units}^2$

20) $x=|y|$ and $2x=−y^2+2$

For exercises 21 - 29, 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.

21) $x=y^4$ and $x=y^5$

22) $y=xe^x,\quad y=e^x,\quad x=0$, and $x=1$.

Answer

$This figure is has two graphs. They are the equations y=xe^x and y=e^x. The graphs intersect, forming a region in between them in the first quadrant.$

$(e−2)\, \text{units}^2$

23) $y=x^6$ and $y=x^4$

24) $x=y^3+2y^2+1$ and $x=−y^2+1$

Answer

$Two curves plotted against X and Y axes. One curve is a sideways parabola opening to the left and labeled 'X equals negative Y squared plus 1.' The other curve looks like a horizontally stretched letter S and is labeled 'X equals Y cubed plus 2 Y squared plus 1.' The apex of the parabola touches the upper curve of the S near X equals 1 and Y equals 0. The lower curve of the S also crosses the parabola near X equals negative 8 and Y equals negative 3.$

$\dfrac{27}{4}\, \text{units}^2$

25) $ y=|x|$ and $ y=x^2−1$

26) $y=4−3x$ and $y=\dfrac{1}{x}$

Answer

$This figure is has two graphs. They are the equations y=4-3x and y=1/x. The graphs intersect, having region between them shaded. The region is in the first quadrant.$

$\left(\dfrac{4}{3}−\ln(3)\right)\, \text{units}^2$

27) $y=x^2−3x+2$ and $ y=x^3−2x^2−x+2$

Answer: $This figure is has two graphs. They are the equations y=x^2-3x+2 and y=x^3-2x^2-x+2. The graphs intersect, having region between them shaded.$
$\dfrac{1}{2}$ square units

28) $y+y^3=x$ and $2y=x$

Answer

$This figure is has two graphs. They are the equations 2y=x and y+y^3=x. The graphs intersect, forming two regions. The regions are shaded.$

$\dfrac{1}{2}$ square units

29) $ y=\sqrt{1−x^2}$ and $y=x^2−1$

For exercises 30 - 37, 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.

30) [T] $x=e^y$ and $y=x−2$

31) [T] $y=x^2$ and $y=\sqrt{1−x^2}$

Answer: $1.067$ square units

32) [T] $y=3x^2+8x+9$ and $3y=x+24$

33) [T] $x=\sqrt{4−y^2}$ and $ y^2=1+x^2$

Answer: $0.852$ square units

34) [T] $x^2=y^3$ and $x=3y$

35) [T] $y=\sqrt{1−x^2}$ and $y^2=x^2$

36) [T] $y=\sqrt{1−x^2}$ and $y=x^2+2x+1$

Answer: $\dfrac{3π−4}{12}$ square units

37) [T] $x=4−y^2$ and $ x=1+3y+y^2$

38) The largest triangle with a base on the $x$-axis that fits inside the upper half of the unit circle $y^2+x^2=1$ is given by $ y=1+x$ and $ y=1−x$. See the following figure. What is the area inside the semicircle but outside the triangle?

$This figure is has the graph of a circle with center at the origin and radius of 1. There is a triangle inscribed with base on the x-axis from -1 to 1 and the third corner at the point y=1.$

39) A factory selling cell phones has a marginal cost function $C(x)=0.01x^2−3x+229$, where $x$ represents the number of cell phones, and a marginal revenue function given by $R(x)=429−2x.$ 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

40) An amusement park has a marginal cost function $C(x)=1000e−x+5$, where $x$ represents the number of tickets sold, and a marginal revenue function given by $R(x)=60−0.1x$. Find the total profit generated when selling $550$ tickets. Use a calculator to determine intersection points, if necessary, to two decimal places.

For exercises 41 - 43, 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?

41) $y=x^2+2x+1$ and $y=−x^2−3x+4$

Answer: $\dfrac{343}{24}$ square units

42) $y=x^4$ and $x=y^5$

43) $x=y^2−2$ and $x=2y$

Answer: $4\sqrt{3}$ square units

For exercises 44 - 45, solve using calculus, then check your answer with geometry.

44) 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?

$This figure is the graph of a circle centered at the origin with radius of 1. There is a circumscribed square around the circle.$

45) Find the area between the perimeter of the unit circle and the triangle created from $y=2x+1,\,y=1−2x$ and $y=−\dfrac{3}{5}$, as seen in the following figure. Is there a way to solve this without using calculus?

$This figure is the graph of a circle centered at the origin with radius of 1. There are three lines intersecting the circle. The lines intersect the circle at three points to form a triangle within the circle.$

Answer: $ \left(π−\dfrac{32}{25}\right)$ 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.

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