Arc Length and Surface Area
Find the exact length of the curve:
y=1+6x^(3/2), 0<=x<=1
2/243(82sqrt(82)-1)
Solve the following integral:
int sin^2(x)cos^3(x)dx
1/3sin^3(x)-1/5sin^5(x)+C
Solve the following integral:
intx^4/(x-1)dx
1/4x^4+1/3x^3+1/2x^2+x+ln|x-1|+C
Why is the following integral improper?
int_1^2x/(x-1)dx
Infinite Discontinuity at x=1
Is the following differential equation linear?
dy/dt=ty
Yes
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)
Solve the following integral:
int xcos^2(x)dx
1/4x^2+1/4xsin(2x)+1/8cos(2x)+C
Solve the following integral:
int (3t-2)/(t+1)dt
3t-5ln|t+1|+C
Why is the following integral improper?
int_-oo^oo x^2e^(-x^2)dx
Infinite Interval of Integration
Solve the following initial-value problem with the initial condition y0=0.
dy/dt=-y-1
y=1+Ce^-t
Find the exact length of the curve:
y^2=4(x+4)^3,0<=x<=2,y>0
2/27(55sqrt55-37sqrt37)
Solve the following integral:
int tan^2xdx
tan(x)-x+C
Solve the following integral:
int(x+4)/(x^2+2x+5)dx
1/2ln(x^2+2x+5)+3/2arctan((x+1)/2)+C
Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
int_0^oo 1/(root4 (1+x))dx
Divergent (oo)
Find the general solution to the differential equation.
x^2y'=(x+1)y
y=Cxe^(-1/x
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)
Solve the following integral:
int_0^(pi/2) sin^5xdx
8/15
Solve the following integral:
int (x^5+x-1)/(x^3+1)dx
1/3x^3-ln|x+1|+C
Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
int_-oo^6 re^(r/3)dr
Convergent (9e^2)
Solve the following differential equation by using integrating factors.
y'=3x+xy
y=Ce^(x^2/2)-3
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
Solve the following integral:
int_0^pi cos^6(theta)d theta
(5pi)/16
Solve the following integral:
int_0^1 (x)/(x^2+4x+13)dx
1/2ln(18/13)-pi/6+2/3arctan(2/3)
Determine whether each integral is convergent or divergent. Evaluate those that are convergent.
int_0^1 ln(x)/sqrtxdx
Convergent (0)
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)