MATH-161 Final

Integration

100

∫(5x-1)/ (x+1)(x-2)

2ln|x+1| + 3ln|x-2| + c

200

∫xln(x)dx

1/2 x² ln(x) - 1/4 x² + c

300

∫sin²(x)cos³(x)dx

1/3 sin³(x) - 1/5 sin⁵(x) + c

400

∫√(x²-1) / x dx

tan(sec^-1(x)) - sec^-1(x) + c

500

∫sec²(x)tan(x)e^tan(x)dx

tan(x)e^tan(x) - e^tan(x) + c

Work, Area of Thin Plates, Hydrostatic Pressure, Fluid Tanks, Surface Area

100

Set up but do not solve an integral for the surface area of y = sin x where 0 ≤ x ≤ π/2 rotated about the x-axis.

∫₀^2π 2πsinx√ (1+cos²x)  dx

200

Suppose that a thin plate with density p = x kg/m², is shaped like the region bounded by the curves y=x² +2 and y=x+2. Find the mass of this plate.

1/12 kg

300

A spring has a natural length of 20 cm. A 40 N force is required to stretch the spring to a length of 30 cm. How much work is done in stretching the spring from 35 cm to 38 cm?

1.98 J

400

Suppose a thin rectangular plate shaped like the region bound by the curves y = 0, y = 3, x = 2, and x = 4 is standing at the bottom of a lake with a depth of 12 meters. Suppose the water has a density of 1000 kg/m²s². Set up, but do not solve, the hydrostatic pressure on one side of this plate.

∫₀³ (1000)(12-y)(2)dy

500

Suppose a fluid with a density of 500 N/m³ is being pumped out of a tank that looks like the region obtained by rotating the curve y = x² for 0 ≤ x ≤ 1 about the y-axis. If the tank is initially only filled up to the height of ¾ m, how much work is done by emptying the tank? (Set up an integral but you do not have to solve.)

 ∫₀^3/4 (πy)(500)(1-y)dy

Disks/Shells

100

What is the formula for disks along a vertical axis?

Disks: ∫π(R² - r²)dy

200

What is the formula for shells along a vertical axis?

Shells: 2π∫rh dy or 2π∫ r(f(x)-g(x))dy

300

Using the methods of disks, set up an integral to find the volume of the solid that results when the region enclosed by the curves is revolved about the x-axis:

y= -x² + 1

y=0

π∫_-1¹ (-x²+1)²dx

400

Use the shell method to write an integral for the volume of the solid generated by revolving the shaded region about the indicated axis. 

2π∫^{√ 3}₀ y(√3 - (3-y²))dy

500

Using the methods of disks, write an integral for the volume of the solid that results, when the region enclosed by the curves, is revolved about the x-axis:

x=-y² + 2

x=y

Axis: x = -2

π∫_-2¹ (-y²+4)² - (y+2)² dy

Series

100

(n!)²/ (n+4)!(n-2)!

n(n-1) / (n+4)(n+3)(n+2)(n+1)

200

∑⁴⁰⁰_{n=1 6(4/9)}ⁿ

(6-6(4/9)⁴⁰¹)/(1- 4/9

300

∑^∞ _n=1 ((n!)/(2n)!)ⁿ

converges by root test

400

∑^∞ _n=1 (n+1)/(n³ + n)

= 1

converges by limit comparison test

500

∑^∞_n=1 (n²/eⁿ)

1/e

converge by the ratio test

Taylor Series and Complex Numbers

100

When u = 3 - 4i and b = 5 - i … 

Solve for: ū + ib

4 + 9i

200

Suppose the Taylor series for

 f(x) = 5 + 4x + x² - 6x³ + 8x⁴ + ... 

What are the first three terms of the Taylor series for xf''(x⁴)?

2x - 36x⁵ + 96x⁹

300

Rewrite the complex numbers in polar form:

Z = 5 - 5i

z=√32 e^{i(7π /4)}

400

Rewrite the complex number in rectangular form. 

z = 3e^πi

z=-3

500

Use Euler's formula to simplify the expression 

e^{3 + (3π /2)i}

-ie³