26 Questions
What is the purpose of the rule of differentiation under the integral sign?
To differentiate an integral with respect to a parameter
What is the formula for $\frac{dI}{dt}$ in the Leibniz integral rule?
$\frac{dI}{dt} = \int_{a(t)}^{b(t)} \frac{\partial f}{\partial t} dx + f(b(t),t) \frac{db}{dt} - f(a(t),t) \frac{da}{dt}$
What is one of the key points of the Leibniz integral rule?
The formula consists of two parts: the integral of the partial derivative of f with respect to t, and the boundary terms
Which of the following is an application of the Leibniz integral rule?
Analyzing economic models with integral equations
What is an important note to consider when applying the Leibniz integral rule?
The rule assumes that the functions involved are sufficiently smooth and continuous
What is possible with the Leibniz integral rule?
Differentiating an integral with respect to multiple parameters
What is tracing of curves typically done in?
Cartesian, Parametric and polar forms
Which of the following is not a form of rectification?
Spherical forms
What is the relation between the rule of differentiation under integral sign and tracing of curves?
One is used to find the derivative of an integral, the other is used to graph equations
What is a common application of rectification in real life?
Architecture
What does the rule of differentiation under integral sign allow us to do?
Differentiate under the integral sign
What is commonly traced in three forms?
Curves
What is the process of finding the length of a curve called?
Rectification
In which forms is rectification of curves typically done?
Cartesian, Parametric, and polar forms
What is the relation between tracing of curves and rectification?
They are both used to find the length of a curve
What is the main difference between tracing of curves and rectification?
Tracing of curves is used to find the equation of a curve, while rectification is used to find the length of a curve
What is the main application of double integration?
To find the volume of a solid
What is the main advantage of changing the order of integration?
It makes the integration easier
What is the purpose of changing variables from Cartesian to polar coordinates?
To simplify the integration
What is the result of evaluating a double integral?
A value
What is the relationship between the limits of integration and the area of a region?
The limits of integration determine the area of the region
What is the application of double integration?
Evaluating the area of a region
What is the advantage of changing the order of integration?
It reduces the complexity of the integral
Why is it useful to change variables from Cartesian to polar coordinates?
To simplify the integral
What is the result of evaluating a double integral?
The area of a region
What is the purpose of using double integration to evaluate area?
To evaluate the area of a region
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∂fdx+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∂fdx+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 ofxandt -
a(t)andb(t)are differentiable functions oft -
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
fwith respect tot, 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∂fdx+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
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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