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# 数学代写|微积分代写Calculus Assignment代考|Math1131Q Review: area of a circular sector

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## 数学代写|微积分代写Calculus Assignment代考|Review: area of a circular sector

A sector of a circle is a region in the circle’s interior between two radii, such as the shaded regions in figure 1 ; think “pizza slices.”

The area of a circle is known to be $A=\pi r^2$, where $r$ is the radius of the circle. The sector in the middle of figure 1 covers half the circle’s interior, so its area is $A=\frac{1}{2} \pi r^2$. This proportionality argument works in general. The proportion of a full rotation represented by the angle is the same as the proportion of the circle’s area represented by the sector. Because the proportion of a full rotation is $\frac{\theta}{2 \pi}$, the area of a sector is
$$A=\frac{\theta}{2 \pi} \cdot \pi r^2=\frac{1}{2} r^2 \theta$$
AREA OF A SECTOR OF A CIRCLE
The area of a sector with angle $\theta$ and radius $r$ is
$$A=\frac{1}{2} r^2 \theta .$$

## 数学代写|微积分代写Calculus Assignment代考|Areas in polar coordinates

Suppose we are given a polar curve $r=f(\theta)$ and wish to know the area of a region enclosed by the curve or a portion of the curve, such as shaded in figure 2. The region under a curve $y=f(x)$ stretches “downward” to $y=0$, which is a line. Here, however the regions stretch “inward” to $r=0$, which is a point. Instead of using rectangles, we therefore approximate area using sectors of a circle, as in figure $3 .$

Figure 3 uses $n=5$ sectors to approximate the area inside the curve $r=f(\theta)$ between $\theta=c$ and $\theta=d$. Using $n=10$ sectors, as in figure 4 , appears to reduce the “extra” area and “missing” area, which should make for a better approximation. As you may have anticipated, the idea is to use infinitely many sectors to get the exact area inside the curve.

## 数学代写|微积分代写微积分作业代考|复习:圆扇区面积

$$A=\frac{\theta}{2 \pi} \cdot \pi r^2=\frac{1}{2} r^2 \theta$$

$$A=\frac{1}{2} r^2 \theta .$$

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