Definite Integral of a Piecewise Function

The video tutorial explains how to evaluate a definite integral of a piecewise function f(x), which is defined as x+1 for x<0 and cosine(pi x) for x>=0. The integral is split into two parts: from -1 to 0 and from 0 to 1. The first part involves finding the antiderivative of x+1, resulting in a value of 1/2. The second part involves using u-substitution to find the antiderivative of cosine(pi x), which evaluates to 0. The final result of the definite integral from -1 to 1 is 1/2.

What is the piecewise definition of f(x) for x < 0?

• f(x) = x - 1

• f(x) = sine(pi x)

• f(x) = cosine(pi x)

• f(x) = x + 1

Why is it useful to split the integral from -1 to 1 into two parts?

• To simplify the calculation by using different functions for different intervals

• To avoid calculating the integral

• To make the function continuous

• To change the limits of integration

What is the antiderivative of x + 1?

• x^2/2 + x

• x^2/2 + 1

• x^2/2 - x

• x^2 + 1

What is the result of evaluating the integral from -1 to 0 of x + 1?

• -1/2

• 1/2

• 1

• 0

Which technique is used to evaluate the integral of cosine(pi x)?

• Partial fraction decomposition

• U-substitution

• Integration by parts

• Trigonometric substitution

What is the antiderivative of cosine(pi x) using u-substitution?

• 1/pi * sine(pi x)

• sine(pi x)

• cosine(pi x)

• 1/pi * cosine(pi x)

What is the value of the integral from 0 to 1 of cosine(pi x)?

• 1/2

• 1

• 0

• pi

What is the final result of the definite integral from -1 to 1 of f(x)?

• 1/2

• 1

• -1/2

• 0

What is the purpose of evaluating each part of the integral separately?

• To handle the piecewise nature of the function

• To make the function continuous

• To change the limits of integration

• To avoid using trigonometric identities

What is the significance of the point x = 0 in the piecewise function?

• It is the maximum value of the function

• It is where the function changes from x + 1 to cosine(pi x)

• It is the midpoint of the interval

• It is where the function becomes undefined