Exam #2 Review Jeopardy

Arc Length and Surface Area

100

Find the exact length of the curve:

y=1+6x^(3/2), 0<=x<=1

2/243(82sqrt(82)-1)

200

Find the exact area of the surface obtained by rotating the curve about the x-axis:

y=x^3, 0<=x<=2

pi/27(145sqrt145-1)

300

Find the exact length of the curve:

y^2=4(x+4)^3,0<=x<=2,y>0

2/27(55sqrt55-37sqrt37)

400

Find the exact area of the surface obtained by rotating the curve about the x-axis:

x=1+2y^2, 1<=y<=2

pi/24(65sqrt65-17sqrt17)

500

Find the exact length of the curve:

y=1-e^-x,0<=x<=2

ln(1+sqrt(1+e^-4))+2-sqrt(1+e^-4)-ln(1+sqrt2)+sqrt2

Trigonometric Substitution

100

Solve the following integral:

int sin^2(x)cos^3(x)dx

1/3sin^3(x)-1/5sin^5(x)+C

200

Solve the following integral:

int xcos^2(x)dx

1/4x^2+1/4xsin(2x)+1/8cos(2x)+C

300

Solve the following integral:

int tan^2xdx

tan(x)-x+C

400

Solve the following integral:

int_0^(pi/2) sin^5xdx

8/15

500

Solve the following integral:

int_0^pi cos^6(theta)d theta

(5pi)/16

Partial Fractions

100

Solve the following integral:

intx^4/(x-1)dx

1/4x^4+1/3x^3+1/2x^2+x+ln|x-1|+C

200

Solve the following integral:

int (3t-2)/(t+1)dt

3t-5ln|t+1|+C

300

Solve the following integral:

int(x+4)/(x^2+2x+5)dx

1/2ln(x^2+2x+5)+3/2arctan((x+1)/2)+C

400

Solve the following integral:

int (x^5+x-1)/(x^3+1)dx

1/3x^3-ln|x+1|+C

500

Solve the following integral:

int_0^1 (x)/(x^2+4x+13)dx

1/2ln(18/13)-pi/6+2/3arctan(2/3)

Improper Integrals

100

Why is the following integral improper?

int_1^2x/(x-1)dx

Infinite Discontinuity at x=1

200

Why is the following integral improper?

int_-oo^oo x^2e^(-x^2)dx

Infinite Interval of Integration

300

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

int_0^oo 1/(root4 (1+x))dx

Divergent  (oo) 

400

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

int_-oo^6 re^(r/3)dr

Convergent  (9e^2) 

500

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.

int_0^1 ln(x)/sqrtxdx

Convergent  (0) 

Differential Equations

100

Is the following differential equation linear?

dy/dt=ty

 

Yes

200

Solve the following initial-value problem with the initial condition y₀=0.

dy/dt=-y-1

 

y=1+Ce^-t

300

Find the general solution to the differential equation.

x^2y'=(x+1)y

y=Cxe^(-1/x

400

Solve the following differential equation by using integrating factors.

y'=3x+xy

y=Ce^(x^2/2)-3

500

Solve the following initial-value problem by using integrating factors.

(2+x)y'=y+2+x , y(0)=0

y=(x+2)ln((x+2)/2)