Examples of area in the following topics:
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- Area is a quantity that expresses the extent of a two-dimensional surface or shape, or planar lamina, in the plane.
- Find the area between the two curves $f(x)=x$ and $f(x)= 0.5 \cdot x^2$ over the interval from $x=0$ to $x=2$.
- Since $x > 0.5 \cdot x^2$ over the interval from $x=0$ to $x=2$, the area can be calculated as follows:
- The area between two graphs can be evaluated by calculating the difference between the integrals of the two functions.
- Evaluate the area between two functions using a difference of definite integrals
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- Area and arc length are calculated in polar coordinates by means of integration.
- Then, the area of $R$ is:
- The area of each constructed sector is therefore equal to
- And the total area is the sum of these sectors.
- The area of the region $R$ can also be calculated by integration.
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- In plane geometry and area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter.
- Green's theorem can be used to compute area by line integral.
- The area is given by $A = \iint_{D}\mathrm{d}A$.
- Provided we choose $L$ and $M$ such that: $\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} = 1$, then the area is given by $A=\oint_{C} x$.
- Possible formulas for the area of $D$ include: $A=\oint_{C} xdy$, $A = -\oint_{C} ydx$, and $A = \frac{1}{2}\oint_{C} (xdy - ydx)$.
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- Infinitesimal calculus provides us general formulas for the arc length of a curve and the surface area of a solid.
- We will also use integration to calculate the surface area of a three-dimensional object.
- For rotations around the $x$- and $y$-axes, surface areas $A_x$ and $A_y$ are given, respectively, as the following:
- Now, calculate the surface area of the solid obtained by rotating $f(x)$ around the $x$-axis:
- Use integration to find the surface area of a solid rotated around an axis and the surface area of a solid rotated around an axis
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- Defined integrals are used in many practical situations that require distance, area, and volume calculations.
- We ask, "What is the area under the function $f$, over the interval from $0$ to $1$?
- " and call this (yet unknown) area the integral of $f$.
- Its area is exactly $1$.
- So the exact value of the area under the curve is computed formally as:
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- We ask, "What is the area under the function $f$, over the interval from 0 to 1?
- " and call this (yet unknown) area the integral of $f$.
- Its area is exactly $1$.
- Summing the areas of these rectangles, we get a better approximation for the sought integral, namely:
- So the exact value of the area under the curve is computed formally as:
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- If the curve is described by the function $y = f(x) (a≤x≤b)$, the area $A_y$ is given by the integral $A_x = 2\pi\int_a^bf(x)\sqrt{1+\left(f'(x)\right)^2} \, dx$ for revolution around the $x$-axis.
- If the curve is described by the parametric functions $x(t)$, $y(t)$, with $t$ ranging over some interval $[a,b]$ and the axis of revolution the $y$-axis, then the area $A_y$ is given by the integral:
- Likewise, when the axis of rotation is the $x$-axis, and provided that $y(t)$ is never negative, the area is given by:
- Its area is therefore:
- Use integration to find the area of a surface of revolution
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- Pressure is given as $p = \frac{F}{A}$ or $p = \frac{dF_n}{dA}$, where $p$ is the pressure, $\mathbf{F}$ is the normal force, and $A$ is the area of the surface on contact.
- Pressure ($p$) is force per unit area applied in a direction perpendicular to the surface of an object.
- While pressure may be measured in any unit of force divided by any unit of area, the SI unit of pressure (the newton per square meter) is called the pascal (Pa).
- Mathematically, $p = \frac{F}{A}$, where $p$ is the pressure, $\mathbf{F}$ is the normal force, and $A$ is the area of the surface on contact.
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- Integration by parts may be thought of as deriving the area of the blue region from the total area and that of the red region.
- The area of the blue region is $A_1=\int_{y_1}^{y_2}x(y)dy$.
- Similarly, the area of the red region is $A_2=\int_{x_1}^{x_2}y(x)dx$.
- The total area, $A_1+A_2$, is equal to the area of the bigger rectangle, $x_2y_2$, minus the area of the smaller one, $x_1y_1$: $\int_{y_1}^{y_2}x(y)dy+\int_{x_1}^{x_2}y(x)dx=\biggl.x_iy_i\biggl|_{i=1}^{i=2}$.
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- If $g(x) = 0$ (e.g. revolving an area between curve and $x$-axis), this reduces to:
- The area of a ring is:
- Summing up all of the areas along the interval gives the total volume.
- If $g(x)=0$ (e.g. revolving an area between curve and $x$-axis), this reduces to:
- Summing up all of the surface areas along the interval gives the total volume.