6.7 The Fundamental Theorem of Calculus and Definite Integrals

    Cards (35)

    • What does the Fundamental Theorem of Calculus (FTC) link together?
      Derivative and area
    • The Power Rule for integration states that xndx=\int x^{n} dx =xn+1n+1+ \frac{x^{n + 1}}{n + 1} +C C, where nn \neq -1
    • The Constant Multiple Rule states that kf(x)dx=\int k \cdot f(x) dx =kf(x)dx k \int f(x) dx
    • What does a definite integral represent geometrically?
      Area under a curve
    • Definite integrals are evaluated using the formula abf(x)dx=\int_{a}^{b} f(x) dx =F(b) F(b) - F(a)
    • Steps to find the area under f(x) = x^{2}</latex> from a=a =0 0 to b=b =2 2
      1️⃣ Find the antiderivative: F(x)=F(x) =x33 \frac{x^{3}}{3}
      2️⃣ Evaluate at the limits: F(2)=F(2) =83 \frac{8}{3}, F(0)=F(0) =0 0
      3️⃣ Subtract: F(2)F(0)=F(2) - F(0) =83 \frac{8}{3}
    • If F(x)=F(x) =axf(t)dt \int_{a}^{x} f(t) dt, then F(x)=F'(x) =f(x) f(x)
    • What is Part 2 of the Fundamental Theorem of Calculus used to evaluate?
      Definite integrals
    • The Power Rule states that \int x^{n} dx = \frac{x^{n + 1}}{n + 1} +</latex> C
    • If F(x)=F(x) =axf(t)dt \int_{a}^{x} f(t) dt, then F(x)=F'(x) =f(x) f(x)
    • The Fundamental Theorem of Calculus consists of two key parts
    • What does the first part of the Fundamental Theorem of Calculus state about the derivative of \int_{a}^{x} f(t) dt</latex>?
      F(x)=F'(x) =f(x) f(x)
    • F(x)=F(x) =0xt2dt= \int_{0}^{x} t^{2} dt =x33 \frac{x^{3}}{3}
    • If f(x)f(x) is continuous on [a,b][a, b] and F(x)F(x) is any antiderivative of f(x)f(x), then abf(x)dx=\int_{a}^{b} f(x) dx =F(b)F(a) F(b) - F(a) is the statement of the Fundamental Theorem of Calculus, Part 2.
    • Evaluate 02xdx\int_{0}^{2} x dx using the Fundamental Theorem of Calculus.

      2
    • Steps to evaluate definite integrals using the Fundamental Theorem of Calculus
      1️⃣ Find the antiderivative F(x)F(x) of f(x)f(x)
      2️⃣ Evaluate F(b)F(a)F(b) - F(a), where aa and bb are the limits of integration
    • What is the antiderivative of x2x^{2}?

      x33\frac{x^{3}}{3}
    • The value of 13x2dx\int_{1}^{3} x^{2} dx is \frac{26}{3}
    • The Fundamental Theorem of Calculus Part 2 requires finding an antiderivative to evaluate a definite integral.
    • Match the concept with its description:
      Antiderivative ↔️ A function whose derivative equals the original function
      FTC Part 2 ↔️ Method to evaluate definite integrals using antiderivatives
    • What is the antiderivative of xx?

      x22\frac{x^{2}}{2}
    • The value of \int_{0}^{2} x dx</latex> is 2
    • Steps to apply the Power Rule for integration
      1️⃣ Add 1 to the exponent of xx
      2️⃣ Divide by the new exponent
      3️⃣ Add the constant of integration CC
    • The Constant Multiple Rule states that kf(x)dx=\int k \cdot f(x) dx =kf(x)dx k \int f(x) dx.
    • What does a definite integral represent geometrically?
      Area under a curve
    • The formula abf(x)dx=\int_{a}^{b} f(x) dx =F(b)F(a) F(b) - F(a) is used in the Fundamental Theorem of Calculus to evaluate definite integrals.
    • Match the concept with its description:
      Definite Integral ↔️ Area under a curve between two limits
      FTC ↔️ Links derivatives to integrals
    • What is the derivative of axf(t)dt\int_{a}^{x} f(t) dt?

      f(x)</latex>
    • If f(x)f(x) is continuous and F(x)F(x) is an antiderivative of f(x)f(x), then abf(x)dx=\int_{a}^{b} f(x) dx =F(b)F(a) F(b) - F(a) is the Fundamental Theorem of Calculus, Part 2.
    • What is the antiderivative of x^{2}</latex>?
      x33\frac{x^{3}}{3}
    • An antiderivative of f(x)f(x) is a function whose derivative equals f(x)f(x).
    • Match the concept with its formula:
      Antiderivative ↔️ F'(x) = f(x)</latex>
      FTC Part 2 ↔️ abf(x)dx=\int_{a}^{b} f(x) dx =F(b)F(a) F(b) - F(a)
    • The value of 02xdx\int_{0}^{2} x dx is 2
    • What is the formula for finding the area between two curves f(x)f(x) and g(x)g(x)?

      ab(f(x)g(x))dx\int_{a}^{b} (f(x) - g(x)) dx
    • The area between f(x)=f(x) =x2 x^{2} and g(x)=g(x) =x x from 00 to 11 is 16\frac{1}{6}.
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