Fundamental Theorem of Calculus Concepts

This video tutorial explains the Fundamental Theorem of Calculus, which connects differentiation and integration. It provides guidelines for using the theorem to evaluate definite integrals efficiently. The video includes three examples demonstrating the application of the theorem to calculate areas under curves, including functions with non-negative values and those involving square roots.

What does the Fundamental Theorem of Calculus allow us to do with a continuous function on a closed interval?

• Solve differential equations

• Calculate the area of a circle

• Determine the value of the definite integral using the antiderivative

• Find the derivative of the function

When applying the Fundamental Theorem of Calculus, why is it unnecessary to include the constant of integration?

• The constant is always zero

• The constant is irrelevant to the function

• The constant is included in the antiderivative

• The constant cancels out when finding the difference

In the first example, what shape is used to verify the result of the definite integral?

• Circle

• Rectangle

• Triangle

• Trapezoid

What is the value of the definite integral in the first example?

• 8

• 6

• 10

• 12

In the second example, what is the antiderivative of the function -1/2 x^2 + 2x + 2?

• -1/3 x^3 + 2x^2 + 2x

• -1/6 x^3 + 2x^2 + 2x

• -1/6 x^3 + x^2 + 2x

• -1/2 x^3 + x^2 + 2x

What is the area of the shaded region in the second example?

• 21/2 square units

• 10 square units

• 15 square units

• 25/2 square units

In the third example, how is the square root of x expressed using rational exponents?

• x^1/3

• x^1/2

• x^2/3

• x^3/2

What is the antiderivative of x^1/2 in the third example?

• 3/2 x^1/2

• 1/2 x^3/2

• 3/2 x^2/3

• 2/3 x^3/2

What is the final value of the definite integral in the third example?

• 50/3

• 14/3

• 64/3

• 30/3

Why is the definite integral equal to the area of the shaded region in these examples?

• The integrand function is zero

• The integrand function is a constant

• The integrand function is always positive

• The integrand function is non-negative over the interval