Calculus I - Definite Integrals

42 flashcards · Mathematics · by luckydog789 (harrison holbrook)

1. Area functions

   Let f(x)f(x)f(x) be a continuous function, where t≥at \ge at≥a and aaa is fixed The area function (accumulation function) for fff with left endpoint aaa is denoted by A(x)A(x)A(x) A(x)A(x)A(x) gives the net signed area (i.e. positive area - negative area) of the region bounded by the graph of fff and the ttt -axis (the xxx -axis of a typical graph) between t=at=at=a and t=xt=xt=x

2. Area functions as definite integrals

   A(x)=∫axf(t)dtA(x)=\int_a^xf(t)dtA(x)=∫ax​f(t)dt

3. Riemann Sums

   An approximation of an integral by a finite sum of rectangles with width Δx\Delta xΔx and height f(x)f(x)f(x) on the interval Δx\Delta xΔx , either on the left endpoint, middle, or right endpoint in the interval. Let Δxk\Delta x_kΔxk​ be the length of the subinterval [xk−1,xk][x_{k-1},x_k][xk−1​,xk​] (commonly Δx=b−an\Delta x=\frac{b-a}{n}Δx=nb−a​ ) and xk∗x_{k}^*xk∗​ by any point on [xk−1,xk][x_{k-1}, x_k][xk−1​,xk​] for k=1,2,...,nk=1,2,...,nk=1,2,...,n If fff is defined on [a,b][a,b][a,b] th…

4. Riemann Integral

   Let Δxk\Delta x_kΔxk​ be the length of the subinterval [xk−1,xk][x_{k-1},x_k][xk−1​,xk​] (commonly Δx=b−an\Delta x=\frac{b-a}{n}Δx=nb−a​ ) and xk∗x_{k}^*xk∗​ by any point on [xk−1,xk][x_{k-1}, x_k][xk−1​,xk​] for k=1,2,...,nk=1,2,...,nk=1,2,...,n . If fff is defined on [a,b][a,b][a,b] , and all Δx→0\Delta x \to 0Δx→0 , ∫abf(x)dx=lim⁡Δx→0∑k=1nf(xk∗)Δxk\int_a^bf(x)dx=\lim_{\Delta x\to 0} \sum_{k=1}^{n}f(x_{k}^*)\Delta x_k∫ab​f(x)dx=limΔx→0​∑k=1n​f(xk∗​)Δxk​ is the Riemann integral of fff from aaa…

5. Shorthand Notation for definite integrals

   ∫abf(x)dx=∫f(x)dx∣ab\int_a^bf(x)dx=\int f(x)dx \Big|_a^b∫ab​f(x)dx=∫f(x)dx∣ ∣​ab​

6. First Fundamental Theorem of Calculus

   Let A(x)A(x)A(x) be the area function for fff . The First Fundamental Theorem of Calculus states that A′(x)=ddx∫axf(t)dt=f(x)A'(x)=\frac{d}{dx}\int_a^xf(t)dt=f(x)A′(x)=dxd​∫ax​f(t)dt=f(x) Which means A(x)A(x)A(x) is an antiderivative of fff

7. Second Fundamental Theorem of Calculus

   Let F(x)F(x)F(x) be any antiderivative of fff . The Second Fundamental Theorem of Calculus states that ∫abf(x)dx=F(b)−F(a)=F(x)∣ab\int_a^bf(x)dx=F(b)-F(a)=F(x)\Big|_a^b∫ab​f(x)dx=F(b)−F(a)=F(x)∣ ∣​ab​

8. Let fff and ggg be integrable functions on an interval that contains a,ba,ba,b and ppp

   Used in the following flashcards

9. ∫aaf(x)dx\int_a^af(x)dx∫aa​f(x)dx

   ∫aaf(x)dx=0\int_a^af(x)dx=0∫aa​f(x)dx=0

10. ∫baf(x)dx\int_b^af(x)dx∫ba​f(x)dx

    ∫baf(x)dx=−∫abf(x)dx\int_b^af(x)dx=-\int_a^bf(x)dx∫ba​f(x)dx=−∫ab​f(x)dx

11. ∫ab(f(x)±g(x))dx\int_a^b (f(x) \pm g(x))dx∫ab​(f(x)±g(x))dx

    ∫ab(f(x)±g(x))dx=∫abf(x)dx±∫abg(x)dx\int_a^b (f(x) \pm g(x))dx=\int_a^bf(x)dx \pm \int_a^bg(x)dx∫ab​(f(x)±g(x))dx=∫ab​f(x)dx±∫ab​g(x)dx

12. ∫abcf(x)dx\int_a^b cf(x)dx∫ab​cf(x)dx

    ∫abcf(x)dx=c∫abf(x)dx,c∈R\int_a^b cf(x)dx=c\int_a^bf(x)dx, c \in \mathbb{R}∫ab​cf(x)dx=c∫ab​f(x)dx,c∈R

13. ∫abf(x)dx\int_a^b f(x)dx∫ab​f(x)dx

    ∫abf(x)dx=∫apf(x)dx+∫pbf(x)dx\int_a^b f(x)dx=\int_a^pf(x)dx+\int_p^bf(x)dx∫ab​f(x)dx=∫ap​f(x)dx+∫pb​f(x)dx

14. ∫ab∣f(x)∣dx\int_a^b |f(x)|dx∫ab​∣f(x)∣dx

    ∫ab∣f(x)∣dx=\int_a^b |f(x)|dx=∫ab​∣f(x)∣dx= The sum of the areas of the regions bounded by the fff and the xxx -axis on [a,b][a,b][a,b] (i.e. unsigned area change rather than the net signed area change)

15. Let fff and ggg be integrable functions on [a,b][a,b][a,b] , where a"> b>ab>ab>a

    Used in the following flashcards

16. If f(x)≥0f(x) \ge 0f(x)≥0 on [a,b][a,b][a,b]

    ∫abf(x)dx≥0\int_a^b f(x)dx \ge 0∫ab​f(x)dx≥0

17. if f(x)≥g(x)f(x) \ge g(x)f(x)≥g(x) on [a,b][a,b][a,b]

    ∫abf(x)dx≥∫abg(x)dx\int_a^b f(x)dx \ge \int_a^b g(x)dx∫ab​f(x)dx≥∫ab​g(x)dx

18. if m≤f(x)≤Mm \le f(x) \le Mm≤f(x)≤M on [a,b][a,b][a,b] ( fff is bounded on the interval)

    m(b−a)≤∫abf(x)dx≤M(b−a)m(b-a) \le \int_a^bf(x)dx \le M(b-a)m(b−a)≤∫ab​f(x)dx≤M(b−a)

19. Lower sum

    Underestimation of the definite integral of a function. Commonly, this occurs on a continuous function fff on [a,b][a,b][a,b] by (via Riemann sum): Partitioning the interval into nnn non-overlapping sub intervals of width Δx=b−an\Delta x=\frac{b-a}{n}Δx=nb−a​ Finding the absolute minimum in each sub interval, f(ci)f(c_i)f(ci​) finding the sum of the areas of each sub interval, Σi=1nf(ci)Δx\Sigma_{i=1}^{n} f(c_i) \Delta xΣi=1n​f(ci​)Δx As n→∞n \to \inftyn→∞ , the lower sum and the upper sum beco…

20. Upper sum

    Overestimation of the definite integral of a function. Commonly, this occurs on a continuous function fff on [a,b][a,b][a,b] by (via Riemann sum): Partitioning the interval into nnn non-overlapping sub intervals of width Δx=b−an\Delta x=\frac{b-a}{n}Δx=nb−a​ Finding the absolute maximum in each sub interval, f(Ci)f(C_i)f(Ci​) finding the sum of the areas of each sub interval, Σi=1nf(Ci)Δx\Sigma^n_{i=1}f(C_i)\Delta xΣi=1n​f(Ci​)Δx As n→∞n \to \inftyn→∞ , the lower sum and the upper sum become eq…

21. Norm

    (In this situation) the width of the largest subinterval, denoted max Δxi\text{max } \Delta x_imax Δxi​

22. Definite integral notation

    ∫abf(x)dx\int_a^b f(x)dx∫ab​f(x)dx , where aaa is the lower limit of integration, bbb is the upper limit of integration, ∫\int∫ is the integral sign, f(x)f(x)f(x) is the integrand, and dxdxdx is the differential of the independent variable xxx (what is being integrated on)

23. Existence of the definite integral

    If a function fff is continuous on a closed interval [a,b][a,b][a,b] , the then definite integral over the interval exists

24. Mean Value Theorem for Integrals

    If a function fff is a continuous function on a closed interval [a,b][a,b][a,b] , there is some real number uuu , for a≤u≤ba \leq u \leq ba≤u≤b , for which ∫abf(x)dx=f(u)(b−a)\int_a^b f(x)dx=f(u)(b-a)∫ab​f(x)dx=f(u)(b−a)

25. If fff is an even function, then ∫−aaf(x)dx\int_{-a}^a f(x)dx∫−aa​f(x)dx

    ∫−aaf(x)dx=2∫0af(x)dx\int_{-a}^a f(x)dx=2\int_{0}^a f(x)dx∫−aa​f(x)dx=2∫0a​f(x)dx

26. If fff is an odd function, then ∫−aaf(x)dx\int_{-a}^a f(x)dx∫−aa​f(x)dx

    ∫−aaf(x)dx=0\int_{-a}^a f(x)dx=0∫−aa​f(x)dx=0

27. ddx∫h(x)g(x)f(t)dt\frac{d}{dx}\int_{h(x)}^{g(x)}f(t)dtdxd​∫h(x)g(x)​f(t)dt

    ddx∫h(x)g(x)f(t)dt=f(g(x))g′(x)−f(h(x))h′(x)\frac{d}{dx}\int_{h(x)}^{g(x)}f(t)dt=f(g(x))g'(x)-f(h(x))h'(x)dxd​∫h(x)g(x)​f(t)dt=f(g(x))g′(x)−f(h(x))h′(x)

28. Right Riemann Sum

    In a Right Riemann Sum (right rule sum), the definite integral over the interval is approximated via the values at the right endpoints of its subintervals. This can be represented in the form of the Riemann integral as ∫abf(x)dx=lim⁡n→∞∑k=1nf(a+kΔx)Δx\int_a^bf(x)dx=\lim_{n \to \infty}\sum_{k=1}^nf(a+k\Delta x)\Delta x∫ab​f(x)dx=limn→∞​∑k=1n​f(a+kΔx)Δx Other forms: Sn=∑k=1nf(a+kΔx)ΔxS_n=\sum_{k=1}^{n}f(a+k\Delta x)\Delta xSn​=∑k=1n​f(a+kΔx)Δx Sn=f(a+Δx)Δx+f(a+2Δx)Δx+f(a+3Δx)Δx,...S_n=f(a+\Delta…

29. Left Riemann Sum

    In a Left Riemann Sum (left rule sum), the definite integral over the interval is approximated via the values at the left endpoints of its subintervals. This can be represented in the form of the Riemann integral as ∫abf(x)dx=lim⁡n→∞∑k=1nf(a+(k−1)Δx)Δx\int_a^b f(x)dx=\lim_{n\to\infty}\sum_{k=1}^{n}f(a+(k-1)\Delta x)\Delta x∫ab​f(x)dx=limn→∞​∑k=1n​f(a+(k−1)Δx)Δx Other forms: Sn=∑k=1nf(a+(k−1)Δx)ΔxS_n=\sum_{k=1}^{n}f(a+(k-1)\Delta x)\Delta xSn​=∑k=1n​f(a+(k−1)Δx)Δx Sn=f(a)Δx+f(a+Δx)Δx+f(a+2Δx)Δx,…

30. Midpoint Riemann Sum

    In a Midpoint Riemann Sum (midpoint rule sum), the definite integral over the interval is approximated via the values at the middle of its subintervals. This can be represented in the form of the Riemann integral as ∫abf(x)dx=lim⁡n→∞∑k=1nf(a+(2(k−1)+1)Δx2)Δx\int_a^b f(x)dx=\lim_{n\to\infty}\sum_{k=1}^{n}f(a+\frac{(2(k-1)+1)\Delta x}{2})\Delta x∫ab​f(x)dx=limn→∞​∑k=1n​f(a+2(2(k−1)+1)Δx​)Δx Other forms: Sn=∑k=1nf(a+(2(k−1)+1)Δx2)ΔxS_n=\sum_{k=1}^{n}f(a+\frac{(2(k-1)+1)\Delta x}{2})\Delta xSn​=∑k=…

31. Trapezoidal Riemann Sum

    In a Trapezoidal Riemann Sum (trapezoidal rule sum), the definite integral over the interval is approximated via trapezoids formed from the two endpoints of their subintervals. This can be represented in the form of the Riemann integral as ∫abf(x)dx=lim⁡n→∞∑k=1nf(a+(k−1)Δx)+f(a+kΔx)2Δx\int_a^bf(x)dx=\lim_{n \to \infty}\sum_{k=1}^n\frac{f(a+(k-1)\Delta x)+f(a+k\Delta x)}{2}\Delta x∫ab​f(x)dx=limn→∞​∑k=1n​2f(a+(k−1)Δx)+f(a+kΔx)​Δx Other forms: Sn=∑k=1nΔx2(f(a+(k−1)Δx)+f(a+kΔx))S_n=\sum_{k=1}^{n}\…

32. Trapezoidal sum (conceptual definition)

    A trapezoidal sum is the average of a left Riemann sum and a right Riemann sum

33. Error in the midpoint Riemann sum

    Error≤(b−a)3M24n2\text{Error} \leq \frac{(b-a)^3M}{24n^2}Error≤24n2(b−a)3M​ , where MMM is the absolute maximum value of ∣f′′(x)∣|f''(x)|∣f′′(x)∣ on the closed interval [a,b][a,b][a,b]

34. Simpson's Rule (Simpson's Method for approximating definite integrals)

    ∫abf(x)dx=b−a3n[f(x0)+4f(x1)+2f(x2)+4f(x3)...2f(xn−2)+4f(xn−1)+2f(xn)]\int_a^b f(x)dx=\frac{b-a}{3n}[f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)...2f(x_{n-2})+4f(x_{n-1})+2f(x_n)]∫ab​f(x)dx=3nb−a​[f(x0​)+4f(x1​)+2f(x2​)+4f(x3​)...2f(xn−2​)+4f(xn−1​)+2f(xn​)] , where the closed interval [a,b][a,b][a,b] is partitioned into an even number nnn of subintervals [x0,x1],[x1,x2]...[xn−1,xn][x_0, x_1], [x_1,x_2] ... [x_{n-1}, x_n][x0​,x1​],[x1​,x2​]...[xn−1​,xn​] of width b−an\frac{b-a}{n}nb−a​

35. Error in Simpson's Rule

    Error≤(b−a)5M180n4\text{Error} \leq \frac{(b-a)^5M}{180n^4}Error≤180n4(b−a)5M​ , where MMM is the absolute maximum value of ∣f(4)(x)∣|f^{(4)}(x)|∣f(4)(x)∣ on the closed interval [a,b][a,b][a,b]

36. Improper integral (of the first kind)

    If a function fff is continuous on the interval [a,∞)[a,\infty)[a,∞) , then ∫a∞f(x)dx\int_a^{\infty} f(x)dx∫a∞​f(x)dx is called an improper integral and is defined as ∫a∞f(x)dx\int_a^{\infty} f(x)dx∫a∞​f(x)dx =lim⁡b→∞∫abf(x)dx=\lim_{b\to\infty} \int_a^{b} f(x)dx=limb→∞​∫ab​f(x)dx provided the limit exists and is a real number. If this limit exists and is a real number, the improper integral is said to converge. If the limit does not exist or the limit is infinite, the improper integral is said…

37. Evaluating an improper integral

    Anti-differentiate the function Apply the limit to the antiderivative Expand the antiderivative Evaluate the limit of the antiderivative If it is a real number, the integral converges Else, it diverges

38. Both limits of an improper integral (of the first kind) are infinite

    ∫−∞∞f(x)dx=∫−∞cf(x)dx+∫c∞f(x)dx\int_{-\infty}^{\infty}f(x)dx=\int_{-\infty}^cf(x)dx+\int_c^{\infty}f(x)dx∫−∞∞​f(x)dx=∫−∞c​f(x)dx+∫c∞​f(x)dx If both integrals on the right converge, then the left interval converges, and if either (or both) on the right diverge, then the left diverges. (choose some arbitrary value for c that makes evaluating the limits easier)

39. Improper integrals (of the second kind)

    If a function fff is continuous on (a,b](a,b](a,b] , but is not defined at aaa then ∫abf(x)dx\int_a^b f(x)dx∫ab​f(x)dx is an improper integral and ∫abf(x)dx=lim⁡t→a+∫tbf(x)dx\int_a^b f(x)dx=\lim_{t\to a^+}\int_t^b f(x)dx∫ab​f(x)dx=limt→a+​∫tb​f(x)dx provided the limit exists and is a real number. If a function fff is continuous on (a,b](a,b](a,b] but is not defined at bbb then ∫abf(x)dx\int_a^b f(x)dx∫ab​f(x)dx is an improper integral and ∫abf(x)dx=lim⁡t→b−∫atf(x)dx\int_a^b f(x)dx=\lim_{t\to b^…

40. Convergence of ∫1∞1xpdx\int_1^\infty \frac{1}{x^p}dx∫1∞​xp1​dx

    ∫1∞1xpdx\int_1^\infty \frac{1}{x^p}dx∫1∞​xp1​dx converges if 1"> p>1p>1p>1 and diverges if p≤1p \leq 1p≤1

41. Improper integral over an interval where some point is not defined

    If a function fff is continuous on a closed interval [a,b][a,b][a,b] , except at the number ccc in the interval, where fff is not defined, ∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx\int_a^b f(x)dx=\int_a^c f(x)dx+\int_c^bf(x)dx∫ab​f(x)dx=∫ac​f(x)dx+∫cb​f(x)dx provided that both improper intervals on the right converge. If either one of them (or both) don't, then the integral diverges.

42. Comparison test for improper integrals

    Suppose fff and ggg are two functions that are nonnegative and continuous on the interval [a,∞)[a,\infty)[a,∞) and suppose f(x)≥g(x)f(x) \geq g(x)f(x)≥g(x) for all number c"> x>cx>cx>c , where c≥ac \geq ac≥a If ∫a∞f(x)dx\int_a^\infty f(x)dx∫a∞​f(x)dx converges, then ∫a∞g(x)dx\int_a^\infty g(x)dx∫a∞​g(x)dx also converges ∫a∞g(x)dx\int_a^\infty g(x)dx∫a∞​g(x)dx diverges, then ∫a∞f(x)dx\int_a^\infty f(x)dx∫a∞​f(x)dx also diverges