## 数学代写|微积分代写Calculus代写|MTH263

2023年1月5日

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## 数学代写|微积分代写Calculus代写|Trig antiderivatives

As previously stated, all derivative formulas can be reversed to provide antiderivative formulas. An example involving trigonometric functions is the derivative formula
$$\frac{d}{d x} \tan x=\sec ^2 x$$
which yields
$$\int \sec ^2 x d x=\tan x+C$$
When the derivative formula results in the use of a negative sign, a slight adjustment is helpful. Reversing $\frac{d}{d x} \cos x=-\sin x$ gives $\int(-\sin x) d x=\cos x+C$, which is not convenient to use because of the negative sign. If we differentiate negative cosine instead, we get a more convenient form. Reversing
$$\frac{d}{d x}(-\cos x)=\sin x$$
yields the antidifferentiation formula
$$\int \sin x d x=-\cos x+C$$
All six trig derivatives can be reversed in this manner.

The indefinite integral formulas (antiderivative formulas) in the right-most column should be memorized, because each formula is used repeatedly for the remainder of this book.
Example 13 Evaluate the indefinite integral $\int\left(4 \sin x+\sec ^2 x\right) d x$.
Solution Using the formulas for the antiderivatives of $\sin x$ and $\sec ^2 x$, along with the antiderivative sum rule and the antiderivative constant multiple rule, gives
\begin{aligned} \int\left(4 \sin x+\sec ^2 x\right) d x & =4(-\cos x)+\tan x+C \ & =-4 \cos x+\tan x+C . \end{aligned}

## 数学代写|微积分代写Calculus代写|Estimating areas using rectangles

The area of the green-shaded region in figure 1 is commonly described as the area under the curve $y=x^2$ from $x=1$ to $x=5$. When we use this terminology, we need a curve that lies on or above the $x$-axis, and the region is then between the $x$-axis and the curve.

Although the area formulas from geometry do not apply to this region directly, we can use the formulas to attempt to estimate the area. Perhaps the easiest area formula is the one for a rectangle. To use the rectangle area formula, we need to decide how many rectangles to use and where to place them. If we use $n=4$ rectangles of width 1 and let the height of each rectangle match the height of the curve at the upper left corner of the rectangle (figure 2, right), we can then estimate the area under the curve by finding the total area of the four rectangles.

To calculate the total area of the brown rectangles in figure 2, we need to know both the widths and the heights of the rectangles. The width of each rectangle is 1 . Notice that the total width of the region is $5-1=4$ and there are four rectangles. In general, if the region goes from $x=a$ to $x=b$ and there are $n$ rectangles, the width of each rectangle is given by
$$\text { width }=\Delta x=\frac{b-a}{n} .$$
Here, the width of each rectangle is $\Delta x=\frac{b-a}{n}=\frac{5-1}{4}=1$.
We can think of each rectangle as being placed on a subinterval of the region’s interval. In figure 2, left, the green region’s interval along the $x$-axis is the interval from $x=1$ to $x=5$, or the interval $[1,5 \mid$. The brown rectangles in figure 2, right, are placed on the subintervals $[1,2],[2,3],[3,4]$, and $[4,5]$. The heights of the rectangles then match the heights of the curve at the left-hand endpoint of each subintervalthat is, at $x=1, x=2, x=3$, and $x=4$. Because the curve is $y=x^2$, the heights of the rectangles are $y=1^2=1, y=2^2=4, y=3^2=9$, and $y=4^2=16$ (figure 3 ).

We are finally ready to calculate the total area of the brown rectangles. Using the formula $A=w h$ for the area of a rectangle, the areas of the rectangles are $1 \cdot 1,1 \cdot 4,1 \cdot 9$, and $1 \cdot 16$. The total area is
$$1 \cdot 1+1 \cdot 4+1 \cdot 9+1 \cdot 16=30 \text { units }^2$$

# 微积分代考

## 数学代写|微积分代写Calculus代写|Trig antiderivatives

$$\frac{d}{d x} \tan x=\sec ^2 x$$

$$\int \sec ^2 x d x=\tan x+C$$

$$\frac{d}{d x}(-\cos x)=\sin x$$

$$\int \sin x d x=-\cos x+C$$

$$\int\left(4 \sin x+\sec ^2 x\right) d x=4(-\cos x)+\tan x+C$$

## 数学代写|微积分代写Calculus代写|Estimating areas using rectangles

$$\text { width }=\Delta x=\frac{b-a}{n} .$$

$$1 \cdot 1+1 \cdot 4+1 \cdot 9+1 \cdot 16=30 \text { units }^2$$

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## MATLAB代写

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