Test 5

Arc Length

100

What is the integral of the arc length? 

s= ∫ √1+[f'(x)]² dx

200

Set up, but do not evaluate, an integral for the length of the curve.

x=9sin(y), 0 ≤ y≤ π /2

∫_0^pi/2 √1+81cos²(y) dy

300

Set up the integral:

y = 2/3x^3/2 , 0 ≤ x ≤ 6 

L = ∫_0^6 √ 1+x dx

400

Find the exact length of the curve.

L = ∫_0^6 √ 1+x dx

(14/3)√7 - 2/3

500

Find the exact length of the curve.

y= (2/3)x^3/2, 0 ≤ x ≤ 4

2/3(5√ 5 - 1)

Area, Surface of Rev

100

What is the formula for surface of revolution

S = 2π∫ r(x)√1+[f'(x)]² dx

200

Set up the integral for the area of the surface.

xy=5y²-1, 1≤ y≤ 4, x-axis

∫_1^4 2(pi)y√1+(5+y^-2)² dy

300

The given curve is rotated about the x-axis. Set up the integral with respect to x.

x=ln(6y+1), 0≤ y≤ 1

∫_0^ln(7) (pi)(1/3e^x-1/3)√1+1/36e^2x dx

400

The given curve is rotated about the x-axis. Set up the integral with respect to y.

x=ln(6y+1), 0≤ y≤ 1

S= ∫_0^1 2(pi)y√ 1+36/(6y+1)² dy

Para-metrics

100

What is arc length in parametric form?

L = ∫ √ (dx/dt)² + (dy/dt)² dt

200

What is surface area in parametric form?

S= ∫ 2πx √ (dx/dt)² + (dy/dt)² dt

300

Find the points (x,y) corresponding to the parameter values t=-2,0,2.

x=9t²+9t, y=3^t+1

t=-2 -> (18,1/3)

t=0 -> (0,3)

t=2 -> (54,27)

400

Find (dx/dt), (dy/dt), (dy,dx).

x=6t³+4t, y=3t-5t²

(dx/dt) = 18t²+4

(dy/dt) = 3-10t

(dy/dx) = (3-10t)/(18t²+4)

500

Find the surface area generated by rotating the given curve about the y=axis.

x=6t^2, y=4t³, 0≤ t≤ 1

*HINT: u=1+t² & du=2tdt*

24(pi)((4(√2)/5 + 4/5)

Polar Coordinates

100

P(x,y) = P(___, ___)

P(r, θ) = P(___, ___)

P(x, y) = P(rcosθ , rsinθ )

P(r, θ ) = P(√x²+y², arctan(y/x))

200

What is the area under a curve (polar)?

A= 1/2∫ r²dθ 

300

Plot the point whose polar coordinates are given. Then find two other pairs of polar coordinates of this point, one with r>0 and one with r<0.

(8, pi/4)

r>0 : (8, 9pi/4)

r<0 : (-8, 5pi/4)

400

Consider the following curve:

r²cos(2θ )=64

Find the Cartesian equation for the curve.

x²-y²=64

500

Find a polar equation for the curve represented by the given Cartesian equation.

x=-4

r=-4secθ 

Random

100

What year did the first Thanksgiving take place?

1621

200

What state in the USA consumes the most turkey on Thanksgiving?

California

300

What president made Thanksgiving a national holiday?

Abraham Lincoln

400

What was the most popular Thanksgiving travel destination?

Orlando, FL

500

What household hazard triples on Thanksgiving?

Fires