# Wednesday 30 August class

Topics: (may be updated when I have time)

• Antiderivatives problem #18 on WeBWorK

• Definite integrals and the Fundamental Theorem of Calculus

Note: there are two different things which have very similar notations, even though they are not the same type of object at all. They are:

The indefinite integral: this represents the set of all of the antiderivatives of the function. (Recall that an antiderivative of $f(x)$ is a function whose derivative is $f(x)$). The indefinite integral is a function containing an arbitrary, undetermined constant (the constant of integration).

Notation for the indefinite integral: $\int f(x)\textrm{d}x$

A definite integral: this represents a signed area or a sum of signed areas, the areas between the graph of &latex f(x)\$ and the x-axis, between two bounds. A definite integral is generally a number.

Notation for a definite integral: $\int_{a}^{b} f(x)\textrm{d}x$

Even though these two things are unrelated to begin with, the Fundamental Theorem of Calculus (part II) comes in to tell us that we can evaluate a definite integral by using antiderivatives, so the same type notation is used for both things!

[This is like the fact that $\frac{2}{5}$ represents two different things: it is the quotient when 2 is divided by 5, and it is also the amount we get when we cut a unit length into 5 pieces and take 2 of them. It happens that the second thing is the answer to the first division problem, so we can use the same notation and usually don’t worry about it!]

Here are some videos from Khan Academy:

The Fundamental Theorem of Calculus (part I)

The Fundamental Theorem of Calculus (part II)

A worked example

Homework:

• Make sure that you have done everything from the first day post.

• Review finding definite integrals and the Fundamental Theorem of Calculus.

• Finish the WeBWorK on Antiderivatives and do the new assignments on the Fundamental Theorem of Calculus – due by Tuesday 11 PM, but don’t wait to the last minute!

• There will be a quiz at the start of class next time. Be on time! It will be on finding antiderivatives (and solving differential equations)

• Monday is a holiday. Happy Labor Day! Next class is next Wednesday, the 6th of September.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question.

# WeBWorK assignment “Antiderivatives” problem 18

The problem gives you two values for the function f(x) at two different x-values. (Different students will get different numbers.)

This is OK, you will have to find f(x) by integrating twice and it will have two constants of integration in it. You can then use the two values of f , one at a time, to find the two constants. Hint: Use f(0) first!

If that’s not enough of a hint you can post a question over on Piazza. Don’t forget to tell us exactly what your problem said, as everyone gets a different version of the problem!