{eq}\displaystyle xe^{-x} \int \frac{e^{-2x}}{x^2 e^{-2x}} \ dx {/eq}



{eq}\displaystyle xe^{-x} \int \frac{e^{-2x}}{x^2 e^{-2x}} \ dx {/eq}

Applications of the Integrals

Integration has multiple applications in mathematics and physics. For example:

{eq}\bullet \; {/eq} Area between curves, the area of a region limited by two functions {eq}f(x) {/eq} and {eq}g(x) {/eq} with {eq}f(x) \geq g(x) {/eq}, and the lines {eq}x=a, \; x=b {/eq} equals,

{eq}A=\displaystyle \int_a^b [f(x)-g(x)]\, dx {/eq}.

{eq}\bullet \; {/eq} Volume of bodies, in general, the volume of a body is computed from a triple integral. Nevertheless, if we know the area as a function of the height it takes the simple form,

{eq}V=\displaystyle \int_0^H A(h) \, dh {/eq}.

{eq}\bullet \; {/eq} Work of a force, is a physical quantity associated with the energy variation caused by the application of the force. It is computed from the line integral,

{eq}W=\displaystyle \int_C \vec{F}\cdot d\vec{l} {/eq},

where {eq}\vec{F} {/eq} is the force and {eq}C {/eq} the integration path joining two points.

Answer and Explanation: 1

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The integral yields,

{eq}\begin{align} I&= \displaystyle \int \dfrac{e^{-x}}{x^2 e^{-x}}\\ &= \displaystyle \int...

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Definite Integrals: Definition


Chapter 12 / Lesson 6

A definite integral is found as the limit between a line graphed from an equation, and the x-axis, either positive or negative. Learn how this limit is identified in practical examples of definite integrals.

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