Notes
Average value
The average value of $f$ on $[a, b]$ is
Motion with integrals
- Displacement (change in position) from $t = a$ to $b$: $\displaystyle \int_a^b v(t)\,dt$.
- Total distance traveled: $\displaystyle \int_a^b |v(t)|\,dt$.
- Position: $\displaystyle s(b) = s(a) + \int_a^b v(t)\,dt$.
Accumulation in context
Amount at time $b$ = amount at time $a$ + $\displaystyle \int_a^b (\text{rate in} - \text{rate out})\,dt$. A maximum amount happens where the rate in equals the rate out and changes from greater to smaller (check endpoints too).
Area between curves
- With respect to $x$: $\displaystyle \int_a^b (\text{top} - \text{bottom})\,dx$.
- With respect to $y$: $\displaystyle \int_c^d (\text{right} - \text{left})\,dy$.
- Find intersection points to get the bounds.
Volume
- Known cross sections perpendicular to the $x$-axis: $\displaystyle \int_a^b A(x)\,dx$, where $A(x)$ is the area of one slice. Squares: $A = s^2$; semicircles: $\displaystyle A = \frac{\pi}{8}s^2$ when $s$ is the diameter.
- Disc method around a horizontal axis: $\displaystyle \pi\int_a^b R(x)^2\,dx$.
- Washer method: $\displaystyle \pi\int_a^b \left(R(x)^2 - r(x)^2\right)dx$, where $R$ and $r$ are outer and inner radii from the axis of rotation.
Topics in this unit
Each topic has its own short notes and practice.
- 8.1Finding the Average Value of a Function on an Interval
- 8.2Connecting Position, Velocity, and Acceleration of Functions Using Integrals
- 8.3Using Accumulation Functions and Definite Integrals in Applied Contexts
- 8.4Finding the Area Between Curves Expressed as Functions of x
- 8.5Finding the Area Between Curves Expressed as Functions of y
- 8.6Finding the Area Between Curves That Intersect at More Than Two Points
- 8.7Volumes with Cross Sections: Squares and Rectangles
- 8.8Volumes with Cross Sections: Triangles and Semicircles
- 8.9Volume with Disc Method: Revolving Around the x- or y-Axis
- 8.10Volume with Disc Method: Revolving Around Other Axes
- 8.11Volume with Washer Method: Revolving Around the x- or y-Axis
- 8.12Volume with Washer Method: Revolving Around Other Axes
Practice
$\displaystyle \frac{1}{3}\int_0^3 x^2\,dx = \frac{1}{3} \cdot 9 = 3$.
$\displaystyle \int_0^4 |t - 2|\,dt$ is two triangles of area 2 each.
$\displaystyle \int_0^4 (t - 2)\,dt = 8 - 8 = 0$.
$\displaystyle \int_0^1 (x - x^2)\,dx = \frac{1}{2} - \frac{1}{3}$.
$\displaystyle \pi\int_0^4 (\sqrt x)^2\,dx = \pi\int_0^4 x\,dx = 8\pi$.
Each slice is a square of area $f(x)^2$.
Start amount plus the net accumulated change.
It's a disc problem: the radius is the distance from the curve up to the axis $y = 2$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
Let $R$ be the region bounded by $y = 4 - x^2$ and the $x$-axis.
(a) Find the area of $R$.
Show the answer and scoring guide
Bounds $x = \pm 2$. $\displaystyle \int_{-2}^{2}(4 - x^2)\,dx = \left[4x - \frac{x^3}{3}\right]_{-2}^{2} = \frac{16}{3} - \left(-\frac{16}{3}\right) = \frac{32}{3}$.
Check off the points you earned:
(b) Find the volume when $R$ is rotated about the $x$-axis.
Show the answer and scoring guide
$\displaystyle \pi\int_{-2}^{2}(4 - x^2)^2\,dx = \pi\int_{-2}^{2}(16 - 8x^2 + x^4)\,dx = \pi\left(64 - \frac{128}{3} + \frac{64}{5}\right) = \frac{512\pi}{15} \approx 107.233$.
Check off the points you earned:
(c) $R$ is the base of a solid whose cross sections perpendicular to the $x$-axis are squares. Write an integral for the volume.
Show the answer and scoring guide
$\displaystyle \int_{-2}^{2}(4 - x^2)^2\,dx$, which equals $\displaystyle \frac{512}{15}$.
Check off the points you earned: