Integral Mathematics

What is the definition of a definite integral?

• The definition of a definite integral is the calculation of the area under a curve between two points on the x-axis.

• The definition of a definite integral is the calculation of the average value of a function between two points on the x-axis.

• The definition of a definite integral is the calculation of the area above a curve between two points on the x-axis.

• The definition of a definite integral is the calculation of the slope of a curve between two points on the x-axis.

Evaluate the definite integral: ∫(2x + 3) dx, from x = 1 to x = 5.

• 30

• 20

• 40

• 36

What is the geometric interpretation of a definite integral?

• Length of the curve

• Volume of the solid

• Slope of the curve

• Area under the curve

Evaluate the definite integral: ∫(4x^2 + 2x + 1) dx, from x = 0 to x = 2.

• 8/3

• 10/3

• 14/3

• 46/3

What is the relationship between a definite integral and the area under a curve?

• The definite integral calculates the area under the curve.

• The definite integral calculates the derivative of the curve.

• The definite integral calculates the maximum value of the curve.

• The definite integral calculates the slope of the curve.

Evaluate the definite integral: ∫(sin(x) + cos(x)) dx, from x = 0 to x = π/2.

• 4

• 3

• 2

• 1

What is the fundamental theorem of calculus?

• The fundamental theorem of calculus states that the derivative of a function can be calculated by finding the integral of the function.

• The fundamental theorem of calculus states that the definite integral of a function can be calculated by finding the derivative of the function.

• The fundamental theorem of calculus states that the indefinite integral of a function can be calculated by finding the derivative of the function.

• The fundamental theorem of calculus states that the definite integral of a function can be calculated by finding an antiderivative of the function and evaluating it at the endpoints of the interval.

Evaluate the definite integral: ∫(e^x + 1/x) dx, from x = 1 to x = 2.

• e^2 + ln(2) - e + 2

• e^2 + ln(2) - e

• e^2 + ln(2) - e - 1

• e^2 + ln(2) - e + 1

What is the connection between definite integrals and antiderivatives?

• The definite integral of a function is equal to the antiderivative evaluated at a single point.

• Definite integrals and antiderivatives are unrelated concepts in calculus.

• The definite integral of a function is equal to the derivative of the antiderivative.

• The definite integral of a function is equal to the difference between the antiderivative evaluated at the upper and lower limits of integration.

Evaluate the definite integral: ∫(3x^2 - 2x + 5) dx, from x = -1 to x = 1.

• 12

• 8

• 14

• 10