Calc II Final

Volumes & Applications of Integration

100

Find the volume obtained by rotating the region bounded by y=x, x=0, and x=2 about the x-axis.

V=π∫ 0->2 x² dx=8π/3

200

Which method is usually preferred when rotating a region around a vertical axis and the slices are parallel to the axis of rotation?

Shell Method

300

Set up (do not evaluate) the integral for the volume obtained by rotating the region between y=x² and y=4 about the y-axis using shells.

V=2π∫0->2 x(4−x²) dx

400

A plate is submerged vertically in water. What formula relates pressure to depth?

P=ρgh

500

Find the centroid xˉ of a uniform rod extending from x=0 to x=10

xˉ=5

Advanced Integration

100

∫xe^xdx

xe^x−e^x+C

200

Which technique is most appropriate for

∫xsinxdx? 

Integration by Parts

300

∫sin3xcosxdx

(sin⁴x)/4 +C

400

What trigonometric substitution should be used for

sqrt{9-x²}? 

x=3sinθ 

500

Decompose:

5/[x(x+2)]

5/(2x)−5/[2(x+2)]

Choosing the Right Technique

100

Which integration technique is usually attempted first when the integrand is a rational function?

Partial Fractions (after division if needed)

200

What technique should be used for

∫16−x² dx?

Trigonometric Substitution

300

What technique should be used for

∫ln⁡x dx? 

Integration by Parts

400

What technique should be used for

∫(x+1)/(x²+3x+2) dx?

Partial Fractions

500

Daily Double:

Evaluate

∫1/(xln⁡x) dx

ln∣lnx∣+C

Limits & Inverse Trig

100

Derivative of

arctan(⁡x)

1/(1+x²)

200

Derivative of

arcsin⁡(x)

1/[sqrt(1−x²)]

300

lim x→∞ (lnx)/x

0 

using l'Hôpital's Rule.

400

lim⁡ x→0 (sin⁡x)/x

1

500

lim⁡ x→∞ x/e^x

0

Improper Integrals & Differential Equations

100

What makes an integral improper?

Infinite interval and/or infinite discontinuity.

200

Determine convergence:

∫ 1->∞ 1/x²dx

Converges

300

Determine convergence:

∫1->∞ 1/x dx

Diverges

400

A tank initially contains 100 gallons with 10 lb of salt. Fresh water enters and mixture leaves. What type of differential equation model is this?

Mixing (Mixture) Problem

500

For

∫1->∞ 1/x^pdx

what values of ppp make the integral converge?

p>1