Calculus 3 - Unit 15

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8 Terms

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Fubini’s Theorem (stronger form)

If f(x,y) is continuous throughout a region D then:

a) If D is defined by

D = { (x,y) | a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x) } [type I regions]


b) If D is defined by

D = { (x,y) | h1(y) ≤ x ≤ h2(y), c ≤ y ≤ d } [type II regions]

The final integral should always be a constant.

<p>If f(x,y) is continuous throughout a region D then:</p><p>a) If D is defined by </p><p>D = { (x,y) | a ≤ x ≤ b, g<sub>1</sub>(x) ≤ y ≤ g<sub>2</sub>(x) } [type I regions]</p><p></p><p>b) If D is defined by</p><p>D = { (x,y) | h<sub>1</sub>(y) ≤ x ≤ h<sub>2</sub>(y), c ≤ y ≤ d } [type II regions]</p><p>The final integral should always be a constant.</p>
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<p>Solving double integrals</p>

Solving double integrals

=1

<p>=1</p>
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Double reimann sum

Where delta A = deltaX*deltaY

<p>Where delta A = deltaX*deltaY</p>
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Formula for double integrals (where R = [a,b] x [c,d])

Integral from x=a to x=b (integral from y=c to y=d (f(x,y)dy)dx)

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Average value of f(x,y) over R

1/A * double integral of f(x,y)

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Double integral in polar

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Triple integrals

Use dV = dxdydz

for cylindrical, dV = rdrdθdz

When finding volume, f(x,y,z) = 1

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Average of f(x,y,z) over R

Triple integral/Volume of R