Problems

In Problems 1-43 below, sketch the given curves and find the area of the region bounded by them.

 1            f(x) = 0, g(x) = 5x - x², 0 ≤ x ≤ 4

 2            f{x) =, g(x) = x², 1 ≤ x ≤ 4

 3            , g(x)=l, -1≤x≤1

 4            y = x - 2, y = 3x^1/3, 0 ≤ x ≤ 1 5           

 6           

 7            The x-axis and the curve y = -5 + 6x - x²

 8            The x-axis and the curve y = 1 - x⁴

 9            The y-axis and the curve x = 25 - y²

10            The }?-axis and the curve x = y(8 - y)

11             y = cos x, y = 2cos x, -π/2 ≤ x ≤ π/2

12            y = sin x cos x, y = 1, 0 ≤ x ≤ 7π/2

13            y = -sin x, y = sin x, 0 ≤ x ≤ π

14            y = sin x, y = cos x, 0 < x < π/4

15            y = sin x cos x, y = sin x, 0 < x < n

16            y = sin² x cos x, y = sin x cos x, 0 < x < π/2

17            y = x, y = e^x, 0 ≤ x ≤ 2

18            y = g^-x, y = e^x, 0 ≤ x ≤ 2

19            y = -e^x, y = e^x, -1 ≤x≤ 1

20           y = xe^x2, y = e, 0 ≤ x ≤ 1

21           ,y=l, 0≤x≤2

22           , 0 ≤ x ≤ 2

23           y = 1/x, y = x, 1 < x < 2

24           ,y = ½, 0≤x≤l

25           f(x) = x^3/2, g(x) = x^2/3

26           y = x² - 2x, y = x - 2

27           y = x⁴ - 2x², y = 2x² + 12

28           y = x⁴ - 1, y = x³ - x

29           y = x⁴/(x² + 1), y = l/(x² + 1)

30           , , 0 ≤ x

31            y = 2x², y = x² + 4

32            x = y², x = 2-y²

33            += 1 and the x- and y-axes

34            x²y = 4, x² + y = 5 (first quadrant)

35            y =, y = 2x

36            y = 0, y = x³ + x + 2, x = 2

37           y = 2x + 4, y = 2 - 3x, y = -x

38            y = x² - 1, y = (x - 1)², y = (x + 1)²

39            y =, y = 1, y = 10 - 2x

40            y = x - 2, y = 2 - x, y =

41             y = -x, y =, y = 3x - 2

42            y = - 2, y = x³ + x, x + y = 3

43            y = x², y = 2x^-2, y = 2x^-3 (first quadrant)

44            Find the area of the ellipse x²/a² + y²/b² = 1. Use the fact that the unit circle has area π.

45            Sketch the four-sided region bounded by the lines y = 1, y = x, y = 2x, and y = 6 - x and find its area.

46            Find the number c > 0 such that the region bounded by the curves y = x, y = - 2x, and x = c has area 6.

47            Find the number c > 1 such that the region bounded by the curves y = 1, y = x^-2, and x = c has area 1.

48            Find the number c such that the region bounded by the curves y = x² and y = c has area 36.

49            Find the number c > 0 such that the region bounded by the curves y = x² and y = cx has area 9.

50            Find the value of c between -1 and 2 such that the area of the region bounded by the lines y = -x, y = 2x, and y = 1 + cx is a minimum.

51            Find the value of c such that the line y = c bisects the region bounded by the curves y = x² and y = 1.

52            Find the value of c such that the line y = cx bisects the region bounded by the x-axis and the curve y = x - x².