Areas under simple curves and regions bounded by curves - Question Bank

1. Calculate the area of the region bounded by the curve y = x^2 and the line y = 10 - x^2.
A) 256/3
B) 128/3
C) 512/3
D) 16/3
2. What is the area of the region bounded by the curve y = x^2 and the line y = 9x.
A) 729/6
B) 81/2
C) 9/3
D) 729/3
3. Find the area of the region bounded by the curve y = x^2 and the line y = 9x - 8.
A) 343/6
B) 1/6
C) 1/2
D) 2/3
4. Determine the area of the region bounded by the curve y = x^2 and the line y = 9 - x^2.
A) 128/3
B) 64/3
C) 256/3
D) 8/3
5. What is the area of the region bounded by the curve y = x^2 and the line y = 7x.
A) 343/6
B) 49/2
C) 7/3
D) 343/3
6. Find the area of the region bounded by the curve y = x^2 and the line y = 7x - 6.
A) 125/6
B) 1/3
C) 1/2
D) 2/3
7. Calculate the area of the region bounded by the curve y = x^2 and the line y = 7 - x^2.
A) 64/3
B) 32/3
C) 128/3
D) 4/3
8. What is the area of the region bounded by the curve y = x^2 and the line y = 5x.
A) 125/6
B) 25/2
C) 10/3
D) 5/3
9. Find the area of the region bounded by the curve y = x^2 and the line y = 5x - 4.
A) 1/6
B) 1/3
C) 1/2
D) 2/3
10. Determine the area of the region bounded by the curve y = x^2 and the line y = 6 - x^2.
A) 16/3
B) 32/3
C) 64/3
D) 4/3
11. What is the area of the region bounded by the curve y = x^2 and the line y = 3x.
A) 9/2
B) 27/4
C) 16/3
D) 4/3
12. Find the area of the region bounded by the curve y = x^2 and the line y = 3x - 2.
A) 1/6
B) 1/3
C) 1/2
D) 2/3
13. Calculate the area of the region bounded by the curve y = x^2 and the line y = 8 - x^2.
A) 32/3
B) 16/3
C) 64/3
D) 4/3
14. What is the area of the region bounded by the curve y = x^2 and the line y = 4x?
A) 16/3
B) 32/3
C) 8/3
D) 4/3
15. Find the area of the region bounded by the curve y = x^2 and the line y = 4x - 3.
A) 1/3
B) 1/2
C) 2/3
D) 1
16. Determine the area of the region bounded by the curve y = x^2 and the line y = -x + 2.
A) 9/2
B) 16/3
C) 27/4
D) 4/3
17. What is the area of the region bounded by the curve y = 1/x, the x-axis, and the lines x = e and x = e^2?
A) 1
B) 2
C) ln(2)
D) e
18. Find the area of the region bounded by the curve y = x^2 and the line y = 2 - x^2.
A) 4/3
B) 2/3
C) 8/3
D) 1
19. Calculate the area of the region bounded by the curve y = x^2 - 2x and the x-axis.
A) 4/3
B) 2/3
C) 8/3
D) 1/3
20. What is the area of the region bounded by the curve y = sqrt(a^2 - x^2)?
A) πa^2
B) πa^2/2
C) πa^2/4
D) 2πa^2
21. Find the area of the region bounded by the curve y = x^2 and the line y = x + 2.
A) 4/3
B) 9/2
C) 16/3
D) 27/4
22. Determine the area of the region bounded by the curve y = x^2 and its normal at x = 1.
A) 1/3
B) 1/2
C) 2/3
D) 1
23. What is the area of the region bounded by the curve y = 1/x^2, the x-axis, and the lines x = 1 and x = 2?
A) 1/2
B) 1
C) 3/2
D) 2
24. Find the area of the region bounded by the curve y = |x - 1| and the x-axis from x = 0 to x = 2.
A) 1
B) 2
C) 3
D) 4
25. Calculate the area of the region bounded by the curve y = x^3 and the line y = x.
A) 0
B) 1/4
C) 1/2
D) 1
26. What is the area of the region bounded by the curve y = sec^2(x), the x-axis, and the lines x = 0 and x = π/4?
A) 1
B) sqrt(2)
C) 2
D) ln(2)
27. Find the area of the region bounded by the curve y = x^2 and the line x = y^2.
A) 1/3
B) 1/2
C) 2/3
D) 1
28. Determine the area of the region bounded by the curve y = 1/sqrt(1-x^2) and the x-axis from x = 0 to x = 1.
A) π/2
B) π
C) 1
D) ln(2)
29. What is the area of the region bounded by the curve y = x^2 and the line y = 4?
A) 16/3
B) 32/3
C) 8/3
D) 4/3
30. Find the area of the region bounded by the curve y = sqrt(x), the x-axis, and the line x = 4.
A) 8/3
B) 16/3
C) 4/3
D) 32/3
31. Calculate the area of the region bounded by the curve y = x^3 - x and the x-axis.
A) 0
B) 1/4
C) 1/2
D) 1
32. What is the area of the region bounded by the curve y = 1/(1+x^2) and the x-axis from x = -∞ to x = ∞?
A) π/2
B) π
C) 2π
D) 1
33. Find the area of the region bounded by the curve y = x^2 and the tangent line at x = 1.
A) 1/3
B) 1/2
C) 2/3
D) 1
34. Determine the area of the region enclosed by the cardioid r = a(1 + cos(θ)).
A) 3πa^2/2
B) πa^2
C) 2πa^2
D) 3πa^2
35. What is the area of the region bounded by the curve y = tan(x), the x-axis, and the lines x = 0 and x = π/4?
A) ln(2)
B) ln(sqrt(2))
C) ln(1/2)
D) 1
36. Find the area of the region bounded by the curve y = x(x-1)(x-2) and the x-axis.
A) 0
B) 1/2
C) 1
D) 3/2
37. Calculate the area of the region bounded by the curve y = x^4 and the line y = 1.
A) 2/5
B) 4/5
C) 1/5
D) 3/5
38. What is the area of the region bounded by the hyperbola x^2/a^2 - y^2/b^2 = 1 and its latus rectum?
A) ab/2
B) ab
C) 2ab
D) ab^2
39. Find the area of the region bounded by the curves y = x^2 and y = 2x - x^2.
A) 1/3
B) 1/2
C) 2/3
D) 1
40. Determine the area of the region bounded by the curve y = x^2 - 4 and the x-axis.
A) 16/3
B) 32/3
C) 8/3
D) 4/3
41. What is the area of the region bounded by y = sin(x) and y = cos(x) between x = 0 and x = π/4?
A) sqrt(2) - 1
B) 2 - sqrt(2)
C) sqrt(2)
D) 1
42. Find the area of the region bounded by the curve y = |x| and the line y = 2.
A) 2
B) 4
C) 6
D) 8
43. Calculate the area of the region bounded by the curve y = x^2 and the line y = x.
A) 1/6
B) 1/4
C) 1/3
D) 1/2
44. What is the area of the region bounded by the curve y = sqrt(a^2 - x^2) and the x-axis from x = 0 to x = a?
A) πa^2/4
B) πa^2/2
C) πa^2
D) a^2
45. Find the area of the region bounded by the circle x^2 + y^2 = r^2.
A) πr^2
B) 2πr^2
C) πr
D) 2πr
46. Determine the area of the region bounded by the curve y = 1/x, the x-axis, and the lines x = 1 and x = 2.
A) ln(2)
B) ln(1)
C) ln(3)
D) ln(4)
47. What is the area of the region bounded by the curve y = e^x, the x-axis, and the lines x = 0 and x = 1?
A) e - 1
B) e
C) e + 1
D) 1
48. Find the area of the region bounded by the curve y = x^3, the x-axis, and the lines x = -1 and x = 1.
A) 0
B) 1/4
C) 1/2
D) 1
49. Calculate the area of the region enclosed by the parabola y^2 = 4ax and the line y = mx.
A) 2a^2m^3 / 3
B) 4a^2m^3 / 3
C) 2a^2 / (3m^2)
D) 4a^2 / (3m^2)
50. What is the area of the region bounded by the curve y = sin(x), the x-axis, and the lines x = 0 and x = π?
A) 0
B) 1
C) 2
D) π
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