Title: Math 2250 Chapter 5 (Integration)

Q: [latex]
The notation
\begin{equation*}
\sum_{k=1}^n a_k
\end{equation*}
means
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A: [latex]
the sum of $n$ terms
$$\sum_{k=1}^n a_k = a_1 + a_2 + \dots + a_n.$$
[/latex]

Q: [latex]
The sum
\begin{equation*}
\sum_{k=1}^n k = 1 + 2 + \dots + n =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\sum_{k=1}^n k = 1 + 2 + \dots + n = \frac{n(n+1)}{2}.
\end{equation*}
Note: A good way to remember this (if you get confused about whether it's $(n+1)$ or $(n-1)$ in the formula) is that
\begin{equation*}
1 + 2 = 3 = \frac{2 \cdot 3}{2}.
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\sum_{k=1}^n k^2 = 1 + 2^2 + 3^2 + \dots + n^2.
\end{equation*}
[/latex]
A: [latex]
$$\sum_{k=1}^n k^2 = 1 + 2^2 + 3^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6}.$$
[/latex]

Q: [latex]
\begin{equation*}
\sum_{k=1}^n k^3 = 1 + 2^3 + 3^3 + \dots + n^3.
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\sum_{k=1}^n k^3 = 1 + 2^3 + 3^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}.
\end{equation*}
[/latex]

Q: [latex]
The relationship between
\begin{equation*}
\int_a^b f(x) \, dx \quad \text{and} \quad \int_b^a f(x) \, dx.
\end{equation*}
is
[/latex]
A: [latex]
$$\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx.$$
[/latex]

Q: [latex]
\begin{equation*}
\int_a^a f(x) \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int_a^a f(x) \, dx = 0
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\int_a^b k f(x) \, dx =
\end{equation*}
[/latex]
A: [latex]
$$\int_a^b k f(x) \, dx = k \int_a^b f(x) \, dx.$$
[/latex]

Q: [latex]
\begin{equation*}
\int_a^b f(x) + g(x) \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int_a^b f(x) + g(x) \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx.
\end{equation*}
[/latex]

Q: [latex]
\begin{equation*}
\int_a^b f(x) \, dx + \int_b^c f(x) \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int_a^b f(x) \, dx + \int_b^c f(x) \, dx = \int_a^c f(x) \, dx.
\end{equation*}
[/latex]

Q: [latex]
If $y = f(x)$ is non-negative and integrable over a closed interval $[a,b]$ then the \emph{area under the curve $f(x)$ over $[a,b]$} is given by
[/latex]
A: [latex]
\begin{equation*}
\operatorname{Area} = \int_a^b f(x) \, dx.
\end{equation*}
[/latex]

Q: [latex]
If $f(x)$ is integrable on $[a,b]$, the \emph{average value of $f(x)$ over $[a,b]$} or \emph{mean value of $f(x)$ over $[a,b]$} is
[/latex]
A: [latex]
\begin{equation*}
\operatorname{Mean}(f) = \frac{1}{b-a} \int_a^b f(x) \, dx.
\end{equation*}
[/latex]

Q: [latex]
(The Fundamental Theorem of Calculus). If $f(x)$ is a continuous function on $[a,b]$ then
\begin{equation*}
\frac{d}{dx} \int_a^x f(y) \, dy =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\frac{d}{dx} \int_a^x f(y) \, dy = f(x).
\end{equation*}
Note: You can say this as ``the derivative of the integral is the function'', or ``differentiation is the inverse of integration''.
[/latex]

Q: [latex]
(The Fundamental Theorem of Calculus) If $F(x) $ is an antiderivative of $f(x)$ or equivalently $F'(x) = f(x)$ then
\begin{equation*}
\int_a^b f(x) \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int_a^b f(x) \, dx = F(b) - F(a).
\end{equation*}
Note: You can say this as ``the integral is an antiderivative''.
[/latex]

Q: [latex]
(The $u$-substitution rule) If $u = g(x)$, then
\begin{equation*}
\int f(g(x)) g'(x) \, dx =
\end{equation*}
[/latex]
A: [latex]
\begin{equation*}
\int f(g(x)) g'(x) \, dx = \int f(u) \, du.
\end{equation*}
[/latex]

Q: [latex]
The definition of the definite integral
\begin{equation*}
\int_a^b f(x) \, dx =
\end{equation*}
as a Riemann sum is
[/latex]
A: [latex]
For each $n$, suppose the interval $[a,b]$ is divided into $n$ subintervals by points
\begin{equation*}
a = x_0 < x_1 = x_0 + h < x_2 = x_1 + h < \dots < x_n = b.
\end{equation*}
We can then define the integral by
\begin{equation*}
\int_a^b f(x) \, dx = \lim_{n \rightarrow \infty} \sum_{k=1}^n f(x_k) h
\end{equation*}
The sum on the right is the area of a collection of rectangles approximating the area under the graph of $f(x)$.
[/latex]