Exploring Definite Integrals

What is the definition of a definite integral?

• A definite integral is the total area of a curve without limits.

• A definite integral is the limit of a Riemann sum that calculates the signed area under a curve over a specific interval [a, b].

• A definite integral represents the average value of a function over an interval.

• A definite integral is the derivative of a function evaluated at two points.

Calculate the definite integral of f(x) = 2x from x = 1 to x = 3.

• 8

• 10

• 6

• 4

What is the Fundamental Theorem of Calculus?

• The Fundamental Theorem of Calculus states that integration and differentiation are inverse processes.

• The Fundamental Theorem of Calculus states that all functions are continuous.

• The Fundamental Theorem of Calculus defines the area under a curve as a constant.

• The Fundamental Theorem of Calculus is only applicable to polynomial functions.

Evaluate the integral ∫_0^2 (3x^2 + 2) dx.

• 10

• 8

• 15

• 12

How does the area under a curve relate to definite integrals?

• The area under a curve is unrelated to integrals.

• The area under a curve can only be calculated using derivatives.

• The area under a curve is always zero.

• The area under a curve is equal to the value of the definite integral of the function over that interval.

Find the definite integral of f(x) = sin(x) from x = 0 to x = π.

• 1

• π

• 0

• 2

What is the geometric interpretation of a definite integral?

• The geometric interpretation of a definite integral is the area under the curve of a function between two points on the x-axis.

• The definite integral measures the distance traveled by a moving object.

• The definite integral represents the slope of a function at a point.

• The definite integral calculates the average value of a function over an interval.

Calculate the definite integral of f(x) = e^x from x = 0 to x = 1.

• 1 - e

• e + 1

• e^2 - 1

• e - 1

Explain how to use substitution in evaluating definite integrals.

• Substitution in definite integrals involves changing variables to simplify the integral, adjusting limits, and then evaluating the integral in the new variable.

• Only apply substitution for indefinite integrals.

• Evaluate the integral without changing variables.

• Use integration by parts instead of substitution.

What is the relationship between the limits of integration and the value of the definite integral?

• Changing the limits of integration has no effect on the integral's value.

• The value of the definite integral is always zero regardless of the limits.

• The limits of integration determine the interval for evaluation, directly influencing the value of the definite integral.

• The limits of integration are irrelevant to the definite integral.