Understanding Definite Integrals and Area Under Curves

The video tutorial explains the concept of defining a new function G(x) as the definite integral of g(t) from t=-3 to t=x. It then demonstrates how to evaluate G at specific values, x=4 and x=8, by calculating the areas under the graph of g(t). The tutorial highlights the importance of considering areas above and below the t-axis, which contribute positively and negatively, respectively, to the integral's value. The evaluations show that G(4) and G(8) result in the same value due to the symmetry of the areas involved.

What is the new function G(x) defined as?

• A sum of g(t) from t = -3 to t = x

• A difference of g(t) from t = -3 to t = x

• A product of g(t) from t = -3 to t = x

• A definite integral of g(t) from t = -3 to t = x

What is the first step in evaluating G(4)?

• Finding the derivative of g(t)

• Multiplying g(t) by 4

• Subtracting g(t) at t = 4 from g(t) at t = -3

• Calculating the area under g(t) from t = -3 to t = 4

Why is the area below the t-axis considered negative?

• Because it is above the t-axis

• Because it is below the graph of g(t)

• Because it is above the graph of g(t)

• Because it is below the t-axis

How is the integral from t = -3 to 0 split?

• Into a single integral from t = 0 to 4

• Into a single integral from t = -3 to 4

• Into two separate integrals from t = -3 to 0 and from t = 0 to 4

• Into three separate integrals from t = -3 to 0, from t = 0 to 4, and from t = 4 to 8

What is the area of the triangle with base 3 and height 3?

• 4.5

• 3

• 9

• 6

What is the value of G(4)?

• -3.5

• 4.5

• 0

• 3.5

What additional areas are considered when calculating G(8)?

• The area from t = -3 to 0

• Only the area from t = 6 to 8

• The areas from t = 0 to 6 and from t = 6 to 8

• Only the area from t = 4 to 6

What is the value of G(8)?

• 0

• -3.5

• 3.5

• 4.5

Why do G(4) and G(8) have the same value?

• Because the function g(t) is linear

• Because the integral is constant

• Because the graph is symmetric

• Because the areas added and subtracted are equal

What happens when you move from G(4) to G(8)?

• You add and subtract different areas

• You add and subtract the same areas

• You only subtract areas

• You only add areas