Quiz for Area under Curves

Q1: What is the area under the curve y = x from x = 0 to x = 2?

A. 1 B. 2 C. 4 D. 3

Q2: What is the definite integral ∫₀¹ (x² + 1) dx?

A. 1.5 B. 1.3 C. 1.33 D. 2.0

Q3: Which formula represents the area under y = f(x) between x = a and x = b?

A. ∫ₐᵇ f(x) dx B. ∫ₐᵇ f(x) dy C. ∫ₐᵇ f'(x) dx D. ∫ₐᵇ f(x²) dx

Q4: The area bounded by y = x² and y = x in [0,1] is:

A. 1/3 B. 1/6 C. 1/2 D. 2/3

Q5: If y = |x|, the area between x = -1 and x = 1 is:

A. 0 B. 1 C. 2 D. 1.5

Q6: What is the area under y = cos(x) from 0 to π?

A. 1 B. 2 C. 0 D. π

Q7: Area under y = e^x from 0 to 1 is:

A. e - 1 B. 1 C. e D. e + 1

Q8: Area between y = sin(x) and x-axis in [0, π] is:

A. 1 B. 2 C. 0 D. π

Q9: The curve y = x³ intersects x-axis at:

A. x = 1 B. x = 0 C. x = -1 D. All of these

Q10: If f(x) = 1/x, find area from x = 1 to x = e.

A. 1 B. e C. log(e) D. ln(e)

Q11: The function f(x) = x² is:

A. Increasing only B. Decreasing only C. Neither D. Both

Q12: Find area between x = 0 and x = 3 under y = x².

A. 9 B. 27 C. 9/2 D. 18

Q13: Which of the following is *not* an area?

A. ∫ f(x) dx from a to b B. ∫ |f(x)| dx from a to b C. ∫ f(y) dy from a to b D. ∫ f'(x) dx from a to b

Q14: The area bounded by parabola y² = 4x and the line x = 4 is:

A. 16 B. 32/3 C. 8 D. 64/3

Q15: Which of the following has *no* finite area?

A. y = 1/x² B. y = 1/x C. y = x D. y = e^x

Q16: Area under f(x) = sin²(x) in [0, π] is:

A. π/2 B. π C. 2π D. π/4

Q17: Which integral gives area between two curves y = f(x), y = g(x)?

A. ∫ₐᵇ [f(x) + g(x)] dx B. ∫ₐᵇ [f(x) - g(x)] dx C. ∫ₐᵇ [g(x) - f(x)] dx D. Either B or C

Q18: If f(x) is positive and decreasing, the trapezoidal rule gives:

A. Overestimate B. Exact value C. Underestimate D. Cannot be determined

Q19: What is area under y = tan(x) from x = 0 to x = π/4?

A. 1 B. ln(2) C. π/4 D. 0.5

Q20: Find the area under f(x) = ln(x) from x = 1 to x = e.

A. 1 B. e C. 0 D. ln(e)