Unit 8: Applications of Integration

16 terms. Say what each one means, then open the card to check.

• Average value of a function

  For f continuous on $[a, b]$, the number $\frac{1}{b - a}\int_a^b f(x)\,dx$. It is the height of the rectangle on $[a, b]$ with the same area as the region under f.

  Example. For $f(x) = 2x$ on $[0, 5]$, $\int_0^5 2x\,dx = 25$, so the average value is $\frac{25}{5} = 5$.

• Mean Value Theorem for integrals

  If f is continuous on a closed interval, then at some point c in the interval, f(c) equals the average value of f on that interval.

  Example. $f(x) = 2x$ on $[0, 5]$ has average value 5, and $f(2.5) = 5$, so $c = 2.5$.

• Average rate of change versus average value

  The average rate of change of F on $[a, b]$ is $\frac{F(b) - F(a)}{b - a}$, the average value of $F'$. The average value of F itself averages the outputs of F.

  Example. For $F(t) = t^2$ on $[0, 4]$, the average rate of change is 4, but the average value of F is $\frac{1}{4}\int_0^4 t^2\,dt = \frac{16}{3}$.

• Displacement

  The net change in position, $\int_a^b v(t)\,dt$. Motion backward counts as negative, so it can cancel motion forward.

  Example. A walker who goes 30 meters east and then 12 meters west has a displacement of 18 meters east.

• Total distance traveled

  The full length of the path, $\int_a^b |v(t)|\,dt$. Split the interval where the velocity changes sign and add the sizes of the pieces.

  Example. The walker who goes 30 meters east and then 12 meters west covers a total distance of 42 meters.

• Position from velocity

  The position at time b is the position at time a plus the displacement: $x(b) = x(a) + \int_a^b v(t)\,dt$.

  Example. A runner at the 100-meter mark with velocity 3 meters per second for 20 seconds reaches the 160-meter mark.

• Speed

  The absolute value of velocity. Speed is never negative, and it increases when velocity and acceleration have the same sign.

  Example. A ball with velocity $-6$ meters per second and acceleration $-9.8$ meters per second squared is speeding up.

• Accumulation from a rate

  If $R(t)$ is the rate at which an amount changes, then the amount at time b is the amount at time a plus $\int_a^b R(t)\,dt$. The units are the rate's units times the time units.

  Example. A pool with 3,000 liters that gains water at a net rate of 40 liters per minute for 15 minutes ends with 3,600 liters.

• Net rate

  The rate in minus the rate out. Integrating it gives the change in the stored amount, and the amount is greatest or least where the net rate changes sign or at an endpoint.

  Example. Sand added to a pile at 5 cubic meters per hour while wind removes 2 cubic meters per hour gives a net rate of 3 cubic meters per hour.

• Area between curves

  If $f(x) \ge g(x)$ on $[a, b]$, the area of the region between them is $\int_a^b (f(x) - g(x))\,dx$: top minus bottom.

  Example. Between $y = 4$ and $y = x^2$ the curves meet at $x = \pm 2$, and the area is $\int_{-2}^{2} (4 - x^2)\,dx = \frac{32}{3}$.

• Limits from intersections

  The x-values (or y-values) where two curves meet, found by setting them equal. They are usually the limits of an area integral.

  Example. $y = x + 6$ and $y = x^2$ meet where $x^2 - x - 6 = 0$, at $x = -2$ and $x = 3$.

• Area by slicing in y

  Write each boundary as x in terms of y and integrate right minus left: $\int_c^d (\text{right} - \text{left})\,dy$. It helps when a curve is a sideways parabola.

  Example. The region between $x = y^2$ and $x = 4$ has area $\int_{-2}^{2} (4 - y^2)\,dy = \frac{32}{3}$.

• Known cross sections

  If each slice perpendicular to the x-axis has area $A(x)$, the solid's volume is $\int_a^b A(x)\,dx$. The side or diameter of a slice is the width of the base at x.

  Example. Over a base whose width at x is $\sqrt{x}$ for $0 \le x \le 9$, square slices give a volume of $\int_0^9 x\,dx = 40.5$.

• Disc method

  Revolving the region under a curve about an axis makes slices that are discs of area $\pi R^2$. The volume is $\pi\int_a^b R(x)^2\,dx$, where R is the distance from the axis to the curve.

  Example. Revolving $y = 3$ for $0 \le x \le 2$ about the x-axis gives a cylinder: $\pi\int_0^2 9\,dx = 18\pi$.

• Washer method

  When the region does not touch the axis, each slice is a washer. The volume is $\pi\int_a^b (R(x)^2 - r(x)^2)\,dx$, with R the outer radius and r the inner radius.

  Example. Revolving the band between $y = 1$ and $y = 2$ for $0 \le x \le 3$ about the x-axis gives $\pi\int_0^3 (4 - 1)\,dx = 9\pi$.

• Radius about another line

  About a horizontal line $y = k$, the radius to a curve $y = f(x)$ is the distance $|f(x) - k|$. Find both radii as distances before squaring.

  Example. About $y = 5$, the curve $y = 1$ is at distance 4 and the curve $y = 3$ is at distance 2.