Unit 6: Integration and Accumulation of Change

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

• Riemann sum

  An estimate of the accumulated amount of a quantity: split the interval into pieces, multiply a height (a value of the function) by each width, and add.

  Example. A cyclist riding at 6, 8 and 7 meters per second during three 10-second stretches covers about 10(6 + 8 + 7) = 210 meters.

• Left and right Riemann sums

  A left sum uses the function’s value at the left end of each subinterval for the height; a right sum uses the value at the right end.

  Example. For $f(x) = x$ on $[0, 2]$ with widths of 1, the left sum is $0 + 1 = 1$ and the right sum is $1 + 2 = 3$.

• Midpoint sum

  A Riemann sum whose heights are the function’s values at the middle of each subinterval.

  Example. For $f(x) = 4 - x$ on $[0, 2]$ with two pieces, the midpoint sum is $1(3.5) + 1(2.5) = 6$.

• Trapezoidal sum

  An estimate that replaces each strip with a trapezoid: width times the average of the two end heights, $\frac{f(a) + f(b)}{2}(b - a)$, added over all pieces.

  Example. Speeds of 20 and 30 meters per second at the ends of a 4-second stretch give a trapezoid of 4 times 25, or 100 meters.

• Overestimates and underestimates

  For an increasing function, a left sum is too low and a right sum too high. For a concave-up function, a trapezoidal sum is too high and a midpoint sum too low.

  Example. $e^x$ is increasing and concave up, so on $[0, 1]$ its left and midpoint sums fall below the integral, and its right and trapezoidal sums fall above it.

• Definite integral

  $\int_a^b f(x)\,dx$, the limit of Riemann sums as the widths shrink to 0. It gives the area above the x-axis minus the area below it, between a and b.

  Example. $\int_0^4 (x - 2)\,dx = 0$, because the triangle below the axis on $[0, 2]$ and the one above it on $[2, 4]$ have equal areas.

• Units of a definite integral

  The units of the function times the units of the input. A rate integrated over time gives an amount.

  Example. If $B(t)$ is a bakery’s selling rate in loaves per hour, $\int_0^{8} B(t)\,dt$ is in loaves.

• Accumulation function

  A function of the form $g(x) = \int_a^x f(t)\,dt$, which adds up the signed area under the graph of f from a fixed start a to a moving end x.

  Example. If $f(t) = 2$ for all t, then $g(x) = \int_1^x 2\,dt = 2(x - 1)$.

• Fundamental Theorem, derivative form

  If f is continuous, then $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$. With an upper limit $u(x)$, the chain rule gives $f(u(x))\,u'(x)$.

  Example. $\frac{d}{dx}\int_0^x \sqrt{1 + t^3}\,dt = \sqrt{1 + x^3}$.

• Fundamental Theorem, evaluation form

  If f is continuous on $[a, b]$ and $F' = f$, then $\int_a^b f(x)\,dx = F(b) - F(a)$.

  Example. $\int_0^2 3x^2\,dx = 2^3 - 0^3 = 8$, using the antiderivative $F(x) = x^3$.

• Net change

  Integrating a rate of change gives the total change in the quantity: $\int_a^b Q'(t)\,dt = Q(b) - Q(a)$.

  Example. If a library’s book count changes at B′(t) books per week, the integral of B′ from week 0 to week 4 is how many books it gained, net, in those 4 weeks.

• Properties of definite integrals

  Integrals over joined intervals add, $\int_a^b + \int_b^c = \int_a^c$; swapping the limits changes the sign; and constants factor out.

  Example. If $\int_0^1 f = 2$ and $\int_1^5 f = 6$, then $\int_0^5 f = 8$ and $\int_5^1 f = -6$.

• Antiderivative

  A function F with $F' = f$. Every antiderivative of f is $F(x) + C$ for some constant C.

  Example. $x^2$, $x^2 + 5$ and $x^2 - 1$ are all antiderivatives of $2x$.

• Indefinite integral

  $\int f(x)\,dx = F(x) + C$, the whole family of antiderivatives of f. The constant C must be written.

  Example. $\int \cos x\,dx = \sin x + C$.

• Power rule for antiderivatives

  $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$. For $n = -1$, $\int \frac{1}{x}\,dx = \ln|x| + C$.

  Example. $\int x^{1/2}\,dx = \frac{2}{3}x^{3/2} + C$.

• Initial condition

  A known value of the function, such as f(0) = 2, used to find the constant C and pick one antiderivative out of the family.

  Example. If $f'(x) = 3$ and $f(0) = 2$, then $f(x) = 3x + C$ with $C = 2$, so $f(x) = 3x + 2$.

• Substitution

  Reversing the chain rule: choose u as an inner function whose derivative also appears, replace $u'(x)\,dx$ with $du$, and integrate in u.

  Example. In $\int 3x^2(x^3 + 1)^4\,dx$, let $u = x^3 + 1$, so $du = 3x^2\,dx$ and the integral is $\frac{(x^3 + 1)^5}{5} + C$.

• Changing the limits

  In a definite integral, after substituting u, change the limits to the u-values of the old limits, or switch back to x before evaluating.

  Example. With $u = 3x + 1$, the x-limits 0 and 1 become the u-limits 1 and 4.