Definite integral of piecewise function | AP Calculus AB | Khan Academy

TL;DR

Splitting the definite integral and evaluating each part separately, the overall definite integral is equal to 1/2.

Transcript

• [Voiceover] So we have a f of x right over here and it's defined piecewise for x less than zero, f of x is x plus one, for x is greater than or equal to zero, f of x is cosine of pi x. And we want to evaluate the definite integral from negative one to one of f of x dx. And you might immediately say, well, which of these versions of f of x am I go... Read More

Key Insights

• ❓ Splitting a definite integral can simplify its evaluation by considering the piecewise function separately for each interval.
• ☺️ Evaluating the antiderivative of a function involves incrementing the exponent of x and dividing by the corresponding exponent.
• 😄 The technique of u-substitution can be used to find the antiderivative of certain functions, such as cosine(pi*x).

Questions & Answers

Q: How is the definite integral of the piecewise function evaluated?

The integral is split into two intervals based on where the function switches. The antiderivative is then computed separately for each interval, and the results are added together.

Q: Why is x+1 the function used for the first interval?

The function x+1 is used for the interval from -1 to 0 because that is the piecewise definition of f(x) in that range.

Q: How is the antiderivative of x+1 computed?

The antiderivative of x+1 is found by increasing the exponent of x by one and dividing the resulting term by that exponent. In this case, it becomes (x^2/2 + x).

Q: How is the antiderivative of cosine(pi*x) computed?

The antiderivative of cosine(pix) involves a technique called u-substitution. By setting u = pix, it can be shown that the antiderivative is (1/pi) * sine(pi*x).

Summary & Key Takeaways

• The given function, f(x), is defined piecewise as x+1 for x<0 and cosine(pi*x) for x>=0.

• To evaluate the definite integral from -1 to 1 of f(x), the integral is split into two intervals: from -1 to 0 and from 0 to 1.

• The antiderivative of x+1 is evaluated for the first interval, and the antiderivative of cosine(pi*x) is evaluated for the second interval.