Leibniz Integral Rule
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Questions and Answers

What is the purpose of the rule of differentiation under the integral sign?

  • To differentiate an integral with respect to a parameter (correct)
  • To integrate a function with respect to a parameter
  • To find the area under a curve
  • To solve differential equations

    • What is the formula for $\frac{dI}{dt}$ in the Leibniz integral rule?

  • $\frac{dI}{dt} = \frac{d}{dt} \int_{a(t)}^{b(t)} f(x,t) dx$
  • $\frac{dI}{dt} = \int_{a(t)}^{b(t)} \frac{\partial^2 f}{\partial t^2} dx$
  • $\frac{dI}{dt} = \int_{a(t)}^{b(t)} \frac{\partial f}{\partial t} dx + f(b(t),t) \frac{db}{dt} - f(a(t),t) \frac{da}{dt}$ (correct)
  • $\frac{dI}{dt} = \int_{a(t)}^{b(t)} f(x,t) dx$

    • What is one of the key points of the Leibniz integral rule?

  • The rule can only be applied to one-dimensional integrals
  • The rule is only applicable to physical quantities
  • The formula consists of two parts: the integral of the partial derivative of f with respect to t, and the boundary terms (correct)
  • The rule assumes that the functions involved are not continuous

    • Which of the following is an application of the Leibniz integral rule?

    <p>Analyzing economic models with integral equations</p> Signup and view all the answers

    What is an important note to consider when applying the Leibniz integral rule?

    <p>The rule assumes that the functions involved are sufficiently smooth and continuous</p> Signup and view all the answers

    What is possible with the Leibniz integral rule?

    <p>Differentiating an integral with respect to multiple parameters</p> Signup and view all the answers

    What is tracing of curves typically done in?

    <p>Cartesian, Parametric and polar forms</p> Signup and view all the answers

    Which of the following is not a form of rectification?

    <p>Spherical forms</p> Signup and view all the answers

    What is the relation between the rule of differentiation under integral sign and tracing of curves?

    <p>One is used to find the derivative of an integral, the other is used to graph equations</p> Signup and view all the answers

    What is a common application of rectification in real life?

    <p>Architecture</p> Signup and view all the answers

    What does the rule of differentiation under integral sign allow us to do?

    <p>Differentiate under the integral sign</p> Signup and view all the answers

    What is commonly traced in three forms?

    <p>Curves</p> Signup and view all the answers

    What is the process of finding the length of a curve called?

    <p>Rectification</p> Signup and view all the answers

    In which forms is rectification of curves typically done?

    <p>Cartesian, Parametric, and polar forms</p> Signup and view all the answers

    What is the relation between tracing of curves and rectification?

    <p>They are both used to find the length of a curve</p> Signup and view all the answers

    What is the main difference between tracing of curves and rectification?

    <p>Tracing of curves is used to find the equation of a curve, while rectification is used to find the length of a curve</p> Signup and view all the answers

    What is the main application of double integration?

    <p>To find the volume of a solid</p> Signup and view all the answers

    What is the main advantage of changing the order of integration?

    <p>It makes the integration easier</p> Signup and view all the answers

    What is the purpose of changing variables from Cartesian to polar coordinates?

    <p>To simplify the integration</p> Signup and view all the answers

    What is the result of evaluating a double integral?

    <p>A value</p> Signup and view all the answers

    What is the relationship between the limits of integration and the area of a region?

    <p>The limits of integration determine the area of the region</p> Signup and view all the answers

    What is the application of double integration?

    <p>Evaluating the area of a region</p> Signup and view all the answers

    What is the advantage of changing the order of integration?

    <p>It reduces the complexity of the integral</p> Signup and view all the answers

    Why is it useful to change variables from Cartesian to polar coordinates?

    <p>To simplify the integral</p> Signup and view all the answers

    What is the result of evaluating a double integral?

    <p>The area of a region</p> Signup and view all the answers

    What is the purpose of using double integration to evaluate area?

    <p>To evaluate the area of a region</p> Signup and view all the answers

    Study Notes

    Rule of Differentiation under Integral Sign (Leibniz Integral Rule)

    Statement of the Rule

    The rule of differentiation under the integral sign, also known as the Leibniz integral rule, is a mathematical formula that allows us to differentiate an integral with respect to a parameter.

    Mathematical Formula

    Let:

    • f(x,t)f(x,t)f(x,t) be a continuous function of xxx and ttt
    • a(t)a(t)a(t) and b(t)b(t)b(t) be differentiable functions of ttt
    • I(t)=∫a(t)b(t)f(x,t)dxI(t) = \int_{a(t)}^{b(t)} f(x,t) dxI(t)=∫a(t)b(t)​f(x,t)dx

    Then:

    dIdt=∫a(t)b(t)∂f∂tdx+f(b(t),t)dbdt−f(a(t),t)dadt\frac{dI}{dt} = \int_{a(t)}^{b(t)} \frac{\partial f}{\partial t} dx + f(b(t),t) \frac{db}{dt} - f(a(t),t) \frac{da}{dt}dtdI​=∫a(t)b(t)​∂t∂f​dx+f(b(t),t)dtdb​−f(a(t),t)dtda​

    Key Points

    • The rule allows us to differentiate an integral with respect to a parameter.
    • The formula consists of two parts: the integral of the partial derivative of fff with respect to ttt, and the boundary terms.
    • The boundary terms are evaluated at the upper and lower limits of the integral.

    Applications

    • The rule has applications in various fields, including:
      • Physics: calculating the derivative of a physical quantity with respect to a parameter.
      • Engineering: optimizing systems with integral constraints.
      • Economics: analyzing economic models with integral equations.

    Important Notes

    • The rule assumes that the functions involved are sufficiently smooth and continuous.
    • The rule can be extended to higher-dimensional integrals and multiple parameters.

    Rule of Differentiation under Integral Sign (Leibniz Integral Rule)

    Statement of the Rule

    • The rule allows differentiation of an integral with respect to a parameter.
    • It is a mathematical formula that involves continuous and differentiable functions.

    Mathematical Formula

    • The formula is: dIdt=∫a(t)b(t)∂f∂tdx+f(b(t),t)dbdt−f(a(t),t)dadt\frac{dI}{dt} = \int_{a(t)}^{b(t)} \frac{\partial f}{\partial t} dx + f(b(t),t) \frac{db}{dt} - f(a(t),t) \frac{da}{dt}dtdI​=∫a(t)b(t)​∂t∂f​dx+f(b(t),t)dtdb​−f(a(t),t)dtda​
    • The formula consists of two parts: integral of the partial derivative of f with respect to t, and the boundary terms.

    Key Points

    • The boundary terms are evaluated at the upper and lower limits of the integral.
    • The rule assumes that the functions involved are sufficiently smooth and continuous.

    Applications

    • The rule has applications in:
      • Physics: calculating the derivative of a physical quantity with respect to a parameter.
      • Engineering: optimizing systems with integral constraints.
      • Economics: analyzing economic models with integral equations.

    Important Notes

    • The rule can be extended to higher-dimensional integrals and multiple parameters.
    • The rule is useful for solving problems involving integrals and parameters.

    Rule of Differentiation under Integral Sign (Leibniz Integral Rule)

    • The rule allows differentiation of an integral with respect to a parameter.
    • Mathematical formula: • Let f(x,t) be a continuous function of x and t. • Let a(t) and b(t) be differentiable functions of t. • Let I(t) = ∫[a(t)]^[b(t)] f(x,t) dx. • Then: dI/dt = ∫[a(t)]^[b(t)] ∂f/∂t dx + f(b(t),t) db/dt - f(a(t),t) da/dt.

    Key Points

    • The formula consists of two parts: the integral of the partial derivative of f with respect to t, and the boundary terms.
    • The boundary terms are evaluated at the upper and lower limits of the integral.

    Applications

    • Physics: calculating the derivative of a physical quantity with respect to a parameter.
    • Engineering: optimizing systems with integral constraints.
    • Economics: analyzing economic models with integral equations.

    Important Notes

    • The rule assumes that the functions involved are sufficiently smooth and continuous.
    • The rule can be extended to higher-dimensional integrals and multiple parameters.

    Rule of Differentiation under Integral Sign (Leibniz Integral Rule)

    • The rule of differentiation under the integral sign, also known as the Leibniz integral rule, allows us to differentiate an integral with respect to a parameter.

    Mathematical Formula

    • The formula is: dI/dt = ∫[a(t) to b(t)] ∂f/∂t dx + f(b(t),t) db/dt - f(a(t),t) da/dt
    • f(x,t) is a continuous function of x and t
    • a(t) and b(t) are differentiable functions of t
    • I(t) = ∫[a(t) to b(t)] f(x,t) dx

    Key Points

    • The rule allows us to differentiate an integral with respect to a parameter
    • The formula consists of two parts: the integral of the partial derivative of f with respect to t, and the boundary terms
    • The boundary terms are evaluated at the upper and lower limits of the integral

    Applications

    • Physics: calculating the derivative of a physical quantity with respect to a parameter
    • Engineering: optimizing systems with integral constraints
    • Economics: analyzing economic models with integral equations

    Important Notes

    • The rule assumes that the functions involved are sufficiently smooth and continuous
    • The rule can be extended to higher-dimensional integrals and multiple parameters

    Rule of Differentiation under Integral Sign (Leibniz Integral Rule)

    • The rule allows us to differentiate an integral with respect to a parameter.
    • The formula consists of two parts: the integral of the partial derivative of fff with respect to ttt, and the boundary terms.
    • The boundary terms are evaluated at the upper and lower limits of the integral.

    Mathematical Formula

    • Let f(x,t)f(x,t)f(x,t) be a continuous function of xxx and ttt
    • Let a(t)a(t)a(t) and b(t)b(t)b(t) be differentiable functions of ttt
    • Let I(t)=∫a(t)b(t)f(x,t)dxI(t) = \int_{a(t)}^{b(t)} f(x,t) dxI(t)=∫a(t)b(t)​f(x,t)dx
    • Then: dIdt=∫a(t)b(t)∂f∂tdx+f(b(t),t)dbdt−f(a(t),t)dadt\frac{dI}{dt} = \int_{a(t)}^{b(t)} \frac{\partial f}{\partial t} dx + f(b(t),t) \frac{db}{dt} - f(a(t),t) \frac{da}{dt}dtdI​=∫a(t)b(t)​∂t∂f​dx+f(b(t),t)dtdb​−f(a(t),t)dtda​

    Applications

    • The rule has applications in physics, engineering, and economics:
      • Physics: calculating the derivative of a physical quantity with respect to a parameter.
      • Engineering: optimizing systems with integral constraints.
      • Economics: analyzing economic models with integral equations.

    Important Notes

    • The rule assumes that the functions involved are sufficiently smooth and continuous.
    • The rule can be extended to higher-dimensional integrals and multiple parameters.

    Double Integrals

    • Evaluation of area by double integration:
      • Cartesian coordinates
      • Change of order of integration
      • Change of variables (Cartesian to polar coordinates)

    Tracing of Curves

    • Tracing of curves in:
      • Cartesian form
      • Parametric form
      • Polar form

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    Understand the Leibniz integral rule, a mathematical formula for differentiating an integral with respect to a parameter. Learn to apply the rule with continuous functions of x and t.

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