# 微积分作业代写calclulus代考| Iterated integration in $\mathbb{R}^{2}$

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## 微积分作业代写calclulus代考|Integration of f : I ⊂ R −→ R

Iterated integration is the workhorse of multiple integrals.
The definition of the multiple integral as the limit of a sum is not practical. Fortunately, there is an alternative. The suggestion is made that we calculate our “volumes” by slicing rather than by dicing.

Consider the thin slice of the “body” under $f$ shown in Figure 4.7. The area of the left-hand side face, that is, the area under the curve of constant $y, y=y_{0}$, is $A\left(y_{0}\right)=\int_{a}^{b} f\left(x, y_{0}\right) \mathrm{d} x$. Similarly, $A\left(y_{0}+\Delta y\right)=\int_{a}^{b} f\left(x, y_{0}+\Delta y\right) \mathrm{d} x$ is the area of the right-hand side face.

If $|\Delta y|$ is a small increment then $A\left(y_{0}\right) \approx A\left(y_{0}+\Delta y\right)$, which is easy to see by expanding $f\left(x, y_{0}+\Delta y\right)$ in a Taylor series about $\left(x, y_{0}\right)$. Then, using the simple two-point trapezoidal rule approximation, the volume of the “slice” is approximately
\begin{aligned} V\left(y_{0}\right) &=\frac{1}{2}\left(A\left(y_{0}\right)+A\left(y_{0}+\Delta y\right)\right) \Delta y \ &=A\left(y_{0}\right) \Delta y+O\left(\Delta y^{2}\right) \end{aligned}

## 微积分作业代写calclulus代考|The geometric interpretation of σn for f : I −→ R

The total volume of the “body” under $f(x, y)$ is then the limiting sum of these $1 \mathrm{D}$ volumes of slices (Definition 4.1) as $\Delta y \rightarrow 0$. That is, the volume under $f(x, y)$ over $R$ is the Riemann integral of $A(y)$ over the interval $c \leq y \leq d$ :
$$V=\int_{c}^{d} A(y) \mathrm{d} y=\int_{c}^{d}\left(\int_{a}^{b} f(x, y) \mathrm{d} x\right) \mathrm{d} y$$
Alternatively, slicing parallel to the $y$-axis instead of the above would give
$$V=\int_{a}^{b} A(x) \mathrm{d} x=\int_{a}^{b}\left(\int_{c}^{d} f(x, y) \mathrm{d} y\right) \mathrm{d} x$$
which must give the exact same value for the volume.
Hence, for integration over the rectangle $[a, b] \times[c, d]$ we have the important result
$$\underbrace{\iint_{R} f(x, y) \mathrm{d} A}{\text {double integral of }}=\underbrace{\int{a}^{b}\left(\int_{c}^{d} f(x, y) \mathrm{d} y\right) \mathrm{d} x=\int_{c}^{d}\left(\int_{a}^{b} f(x, y) \mathrm{d} x\right) \mathrm{d} y}_{\text {iterated integrals of } f \text { over } R}$$
The left-hand side is the definition of a double integral, while the two righthand sides are the actual ways one can evaluate the double integral.

## 微积分作业代写calclulus代考|The geometric interpretation of σn for f : I −→ R

“体”下的总体积F(X,和)那么是这些的极限总和1D切片的体积（定义 4.1）为Δ和→0. 也就是下的音量F(X,和)超过R是黎曼积分一种(和)在区间内C≤和≤d:

$$\underbrace{\iint_{R} f(x, y) \mathrm{d} A} {\text {}}=\underbrace{\int {a}^{b} \left(\int_{c}^{d} f(x, y) \mathrm{d} y\right) \mathrm{d} x=\int_{c}^{d}\left(\int_{a }^{b} f(x, y) \mathrm{d} x\right) \mathrm{d} y}_{\text {迭代积分} f \text { over } R}$$