Volumes & Applications of Integration
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
∫xexdx
xex−ex+C
Which integration technique is usually attempted first when the integrand is a rational function?
Partial Fractions (after division if needed)
Derivative of
arctan(x)
1/(1+x2)
What makes an integral improper?
Infinite interval and/or infinite discontinuity.
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
Which technique is most appropriate for
∫xsinxdx?
Integration by Parts
What technique should be used for
∫16−x2 dx?
Trigonometric Substitution
Derivative of
arcsin(x)
1/[sqrt(1−x2)]
Determine convergence:
∫ 1->∞ 1/x2dx
Converges
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
∫sin3xcosxdx
(sin4x)/4 +C
What technique should be used for
∫lnx dx?
Integration by Parts
lim x→∞ (lnx)/x
0
using l'Hôpital's Rule.
Determine convergence:
∫1->∞ 1/x dx
Diverges
A plate is submerged vertically in water. What formula relates pressure to depth?
P=ρgh
What trigonometric substitution should be used for
sqrt{9-x2}?
x=3sinθ
What technique should be used for
∫(x+1)/(x2+3x+2) dx?
Partial Fractions
lim x→0 (sinx)/x
1
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
Find the centroid xˉ of a uniform rod extending from x=0 to x=10
xˉ=5
Decompose:
5/[x(x+2)]
5/(2x)−5/[2(x+2)]
Daily Double:
Evaluate
∫1/(xlnx) dx
ln∣lnx∣+C
lim x→∞ x/ex
0
For
∫1->∞ 1/xpdx
what values of ppp make the integral converge?
p>1