Definite Integral Evaluation using U-Substitution

This lesson demonstrates how to evaluate a definite integral using u-substitution. It begins by identifying the need for u-substitution due to the integrand's structure. The process involves defining u, solving for dx, and adjusting the limits of integration. The video shows two methods: evaluating the integral in terms of u and in terms of x, both yielding the same result. Finally, it provides a graphical interpretation of the integral, illustrating the area under the curve.

Why is u-substitution necessary for evaluating the given integral?

• Because the integrand is a constant function.

• Because the integrand is the sine of one-half x.

• Because the integrand is the sine of x.

• Because the integrand is a simple polynomial.

What is the differential du in terms of dx for the substitution u = 1/2 x?

• du = 1/2 dx

• du = 2 dx

• du = dx

• du = 3 dx

How do you adjust the limits of integration when using u-substitution?

• By keeping the original limits of x.

• By substituting the x-values into u = 1/2 x.

• By multiplying the limits by 2.

• By adding 1 to each limit.

What is the antiderivative of sine u?

• cosine u

• -cosine u

• sine u

• -sine u

What is the result of the definite integral after evaluating using u-substitution?

• 8

• 2

• 0

• 4

How is the antiderivative expressed when evaluating directly in terms of x?

• 2 cosine (1/2 x)

• -2 cosine (1/2 x)

• -2 cosine x

• 2 sine (1/2 x)

What is the result of the integral when evaluated directly in terms of x?

• 0

• 4

• 8

• 2

Why might textbooks show the antiderivative only in terms of x?

• Because it is more accurate.

• Because it avoids u-substitution.

• Because it is simpler.

• Because it is more complex.

What does the shaded area under the curve represent in the graph?

• The area below the x-axis.

• The derivative of the function.

• The integral of the function.

• The area above the x-axis.

What is the significance of the area being above the x-axis in the graph?

• It indicates an undefined integral value.

• It indicates zero integral value.

• It indicates a positive integral value.

• It indicates a negative integral value.