Problem. Let R be the region z> Va? + y? and a? + y? + 22 < 4. Find the volume of R. Explanation. Let's use spherical again. In this case, to find the volume of R, we need to integrate f(x, y, z) = The region is above a cone and inside a sphere. In spherical the equation of the sphere is If we examine the cross sections, we get that the bounds for p will be ? ? くpS ? Since z is above the cone, o will go from 0 to the cone, i To get ø, notice that the cone z = ²+ y² in spherical has equation o = ? Finally, ? <0< 27. Therefore, the integral becomes 1dV = I f(r(p, 0, $))p² sin(4)dp dø d0 dp do dô
Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
![**Problem.** Let \( R \) be the region \( z \geq \sqrt{x^2 + y^2} \) and \( x^2 + y^2 + z^2 \leq 4 \). Find the volume of \( R \).
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**Explanation.** Let's use spherical coordinates again. In this case, to find the volume of \( R \), we need to integrate \( f(x, y, z) = 1 \). The region is above a cone and inside a sphere. In spherical coordinates, the equation of the sphere is \( \rho = 2 \). If we examine the cross-sections, we get that the bounds for \( \rho \) will be
\[ 0 \leq \rho \leq 2. \]
To get \( \phi \), notice that the cone \( z = \sqrt{x^2 + y^2} \) in spherical coordinates has the equation \( \phi = \frac{\pi}{4}. \) Since \( z \) is above the cone, \( \phi \) will go from 0 to the cone, i.e.,
\[ 0 \leq \phi \leq \frac{\pi}{4}. \]
Finally,
\[ 0 \leq \theta \leq 2\pi. \]
Therefore, the integral becomes
\[
\iiint\limits_R 1 \, dV = \iiint\limits_D f(r(\rho, \theta, \phi)) \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta
\]
\[
= \int_0^{2\pi} \int_0^{\pi/4} \int_0^2 \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta.
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Let us use spherical again. in this case to find the volume of , we need to integrate
The region is above a cone and inside a sphere. In spherical, the equation of the sphere is radius
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