Exam 2

Integral of sin³x

-cosx+(cos³x/3)

How to find the shortest distance from a point to a plane

1. Distance is =sqrt(x-x1)^2+(y-y1)^2+(z-z1)^2 2. Solve the plane equation for z 3. Minimize d^2=(x-x1)^2+(y-y1)^2+(z with respect to x and y 4. Find partial of x 5. Find partial of y 6. Solve partial function with a system of equations aka finding critical points 7. Plug critical point back into distance formula

Integration by u-sub

1. find u 2. find du 3. solve the integral 4. replace u

Integral of sin²x

1/2(x-1/2sin(2x)

Conservative line

By clairauts thm if the partial of P w/respect to y=the partial of Q w/ respect to x

Integral of 1/x

ln(x)

integral of sec^2x dx

tanx + C

cylindrical coordinate substitution

x=rcosθ y=rsinθ r²=x²+y² tanθ=x/y z=z

Spherical Coordinate subsitution

x=µsinσcosθ y=µsinσsinθ z=µcosσ µ²=x²+y²+z² µ²sinσdµdθdσ=dxdydz

Integration by parts

∫fg'=fg-∫f'g

Fundamental Theorem of Line Integrals

∫∨f*dr=f(r(b)-r(a))

How to solve for critical points to find min and max values

1. Find the gradient of f 2. Set the gradient of f=0 and solve, (may need to use a system of equations to solve). 3. Once you have your critical values use the second derivative test to see if they are min, max, or saddle points.

How to find the directional derivative given a point

1. Find the gradient of f 2.Plug in the point to the gradient of f 3. Find the unit vector of the direction 4. dot product the gradient and the unit vector

How to do extreme value problems using Langrange Muiltipliers

1. Find the gradient of the function and the constraint 2. Solve for x.y,z, or lambda 3. Plug values into the constraint and solve 4. Use the solved for values as points in the function to find the max and min values.

How to find the absolute minimum and maximum values of a continuous function f on a closed, bounded set D:

1. Find the values of f at the critical points of f in D 2. Find the extreme values of f on the boundary of D 3.The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.

How to find the function where F=∨f (line integrals)

1. find f(x,y,z) by taking integral of the partial with respect to x.(+g(y,z)) 2. differentiate f(x,y,z) with respect to y. 3. Compare equation to the partial of y to find (gsub(y)(y,z). 4. write the original f(x,y,z) function and then if necessary repeat steps 2-3 but with respect to z 5. Write the complete function

Change of Variable in Multiple integrals

1. find the Jacobian 2. turn functions of x, y into functions of u,v 3. solve for x and y in terms of u and v

Trig Identity for sin²x

1/2(1-cos(2x))

Integral of cos²x

1/2(x+1/2sin(2x)

polar coordinate substitution

r²=x²+y² x=rcosθ y=rsinθ dxdy=rdrdθ