Notes
Antiderivatives
- $F$ is an antiderivative of $f$ if $F' = f$. The indefinite integral $\displaystyle \int f(x)\,dx = F(x) + C$ includes all of them.
- $\displaystyle \int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ ($n \neq -1$), $\displaystyle \int \frac{1}{x}\,dx = \ln|x| + C$, $\displaystyle \int e^x\,dx = e^x + C$, $\displaystyle \int \cos x\,dx = \sin x + C$, $\displaystyle \int \sin x\,dx = -\cos x + C$.
- An initial value problem uses a known point to find $C$.
Riemann sums and the definite integral
- A Riemann sum adds rectangle areas using left, right or midpoint heights, with width $\displaystyle \Delta x = \frac{b - a}{n}$.
- $\displaystyle \int_a^b f(x)\,dx$ is the limit of these sums and equals the net signed area between the graph and the $x$-axis.
Properties
- $\displaystyle \int_b^a = -\int_a^b$, constants factor out, sums split, and adjacent intervals add.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Raise each power by 1 and divide by the new power.
$f = 2x^2 + x + C$ and $f(0) = C = 2$.
Width 1, heights at $x = 0, 1, 2, 3$ are $1, 2, 3, 4$. The sum is 10.
The region is a triangle with base 6 and height 3, so the area is $\displaystyle \frac{1}{2}(6)(3) = 9$.
$\displaystyle \int_7^9 f = \int_2^9 f - \int_2^7 f = 8 - 5 = 3$.
An antiderivative of $\cos x$ is $\sin x$, and of $e^x$ is $e^x$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Let $f(x) = x^2$ on the interval $[0, 2]$. Use 4 equal subintervals, each of width 0.5.
(a) Find the left Riemann sum.
Show the answer and scoring guide
Heights at $x = 0, 0.5, 1, 1.5$: $0, 0.25, 1, 2.25$. Sum of heights is 3.5, so the left sum is $0.5(3.5) = 1.75$.
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(b) Find the right Riemann sum.
Show the answer and scoring guide
Heights at $x = 0.5, 1, 1.5, 2$: $0.25, 1, 2.25, 4$. Sum of heights is 7.5, so the right sum is $0.5(7.5) = 3.75$.
Check off the points you earned:
(c) Explain why $\int_0^2 x^2\,dx$ lies between your two answers, and give an estimate by averaging them.
Show the answer and scoring guide
$x^2$ is increasing on $[0, 2]$, so the left sum is an underestimate and the right sum is an overestimate. The integral is between 1.75 and 3.75. Averaging gives an estimate of $\displaystyle \frac{1.75 + 3.75}{2} = 2.75$.
Check off the points you earned: