MATH-161 Final

Integration

100

∫(5x-1)/ (x+1)(x-2)

-6ln(x+1) + 11ln(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 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 about the x-axis:

y= -x² + 1

y=0

16/15 pi

400

Use the shell method to find the volume of the solid generated by revolving the shaded region about the indicated axis. 

9pi/2

500

Using the methods of disks, find 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

108/5 pi

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 (n²/eⁿ)

1/e

converge by the ratio test

400

∑^∞ _n=1 (n+1)/(n³ + n)

= 1

converges by limit comparison test

500

∑^∞ _n=1 (x-1)ⁿ/n5ⁿ

[-4,6)

(-1)ⁿ/n converges by alt. series test

1/n diverges by nth 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

Approximate (31)^1/3 to three terms. 

3 + 1/27 (31-27) + -2/729 (31-27)² 1/2

400

f(x) = √x  centered at x=16. Estimate √17 to the first three terms.

4 + 1/8 (17-16) + (-1/256) 1/2 (17-9)²

500

Use Euler's formula to simplify the expression 

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

-ie³