18 lines
864 B
Markdown
18 lines
864 B
Markdown
#uw/notes #uw/class/math224
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- definite integral represents the signed area between a function and the x axis between the two bounds
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- $\int_{a}^b f(x) dx$
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- for functions with one variable
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- riemann sums: $$\int_{a}^b f(x) dx = \lim_{ n \to \infty } f(x^*_{k}) \Delta x$$
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- double integral
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- $f(x,y)$
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- $$\int \int_{R} f(x,y) dA$$
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- represents the signed volume between the xy plane and the graph of $z=f(x,y)$ above the region $R$
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- defining region
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- $$R=[a,b] \times [c,d]$$
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- x values are between $a$ and $b$ , y values are between $c$ and $d$
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- slice up solid into equations that can be integrated with one variable
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- $$\int \int_{[a,b] \times [c,d]} f(x,y) dA = \int^b_{a} \int^d_{c}f(x,y)dy dx$$
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- Fubini's theorem
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- $$\int_{a}^b \int_{c}^d f(x,y) dy dx = \int_{c}^d \int_{a}^b f(x,y) dx dy$$
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- can integrate either way (x or y first) |