MATH-161 Final

Integration

100

∫(5x-1)/ (x+1)(x-2)

2ln(x+1) + 3ln(x-2) + c

200

∫sin(x)cos(x)ln(sin(x))dx

(1/2)sin²(x)ln(sin(x)) - (1/4)sin²(x) + 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 and Mass Problems

100

What is joules the unit for?

Energy

200

A spring has a natural length of 20 cm. A 40 N force is required to stretch (and hold 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

300

Suppose a bucket filled with coal, which weighs 4 N, is attached to a 16 m rope which has a density of 2 N/m. If the bucket is dangling off the side of a mineshaft, how much work is required to lift it back to the top?

320 J

400

Find the center of mass for the region bounded by 4 - x² that is in the first quadrant. 

(3/4, 8/5)

500

Find the center of mass for the region bounded by y = 3 - e^-x, x=2 and the y-axis. 

(1.05, 1.29)

Disks/Shells

100

What is the formula for disks along a vertical axis?

Disks: ∫pi(R² - r²)dy

200

What is the formula for shells along a vertical axis?

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

300

Using the methods of disks, find the volume of the solid that results when the region enclosed by the curves is revolved around the axis y = 2. You may leave your answer as an integral.

y = -x² + 3

y = 2

pi ∫¹_-1 (-x²+3)² - (2)²dx 

or

pi ∫¹_-1 (-x²+1)²dx 

400

Using the methods of shells, find the volume of the solid that results when the region enclosed by the curves is revolved around the axis x=-1. You may leave your answer as an integral.

y = -x² + 4

y = x² + 2

x = -1

x= 0

2pi ∫¹ _-1 (x+1)(-x²+4-(x²+2))dx

500

Using the methods of shells, find the volume of the solid that results when the region enclosed by the curves is revolved around the axis x= -4. You may leave your answer as an integral.

y= x² +1

y= 1

x= 2

2pi ∫ ₀²(x+4)(x²)dx

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

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

400

∑^∞_n=1 (n²/eⁿ)

1/e

converges by the ratio test

500

∑^∞ _n=1 (n+1)/(n³ + n)

= 1>0

converges by limit comparison test

Complex Numbers, Polar Coordinates, Terminal Velocity, Exponential Growth and Decay, Euler's Formula

100

When u = 3 - 4i and b = 5 - i … 

Solve for: ū + ib

4 + 9i

200

Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions. r≥ 1, pi≤ θ ≤ 2pi

Look at picture

300

Suppose an object, with a mass of 7 mg, is tossed out of a plane with an initial downward velocity of 4m/s and an initial downward acceleration of 1/5 m/s². Find the terminal velocity of this object. You do not have to simplify your answer. 

√(9.8(7)) / ((-7/16)(1/5 - 9.8))

400

Suppose a quantity y(t) obeys the exponential decay differential equation y' = ky. If y(1) = 9 and y(2) = 5, what must y(0) have been?

A = 5e^2(ln(9/5))

or

A = 9e^2(ln(9/5))

500

Use Euler's formula to simplify the expression 

e^{3 + (3pi/2)i}

-ie³