Midterm 1 Flashcards
Formula for area between two curves
integral of f(x) - g(x) between a and b
OR
integral of f(y) - g(y) between c and d
Volume by general slicing
integral of A(x) between a and b
Volume by disk method
V = integral of (pi * (f(x)^2)) between a and b
OR
V = integral of (pi * (p(y)^2))dx between v and d
Volume by washer method
V = integral of (pi * ( (f(x)^2) -g(x)^2) ) dx between a and b
Volume by shell method
V = integral of 2 * pi * x * ( f(x) - g(x) )dx between a and b
Length of a curve
L = integral of sqrt( 1 + ( f’(x)^2 ) )dx between a and b
Hooke’s Law
F(x) = kx
Work formula
W = integral of F(x)dx between a and b
Work for stretching/compressing
W = integral of kxdx between 0 and x
OR
(1/2)k * x^2
Work for lifting
W = integral of ro * g * A(y) * D(y)dy A(y) = area of slices D(y) = distance slices rise
Force/ Pressure
W = integral of ro * g * (a - y) * w(y)dy
Integration by parts
integral of udv = u * v - integral of v*du
Integration by parts definite integral
integral of ( u(x) * v(x) ) between a and b = u(x) * v(x) between a and b - integral of v(x) * u’(x) between a and b
Integral of lnx dx
xlnx -x + C
Integral of sin^n * x dx
-(sin^(n-1) * xcosx)/n + (n-1)/n * integral of sin^ (n-2) *dx
Integral of cos^n *x dx
(cos^(n-1) *xsinx)/n + (n-1)/n * integral of cos^(n-2) *dx
Integral of tan^n *x dx
(tan^ (n-1) * x)/(n-1) - integral of tan^ (n-2) * xdx, n doesn’t equal 1
Integral of sec^n *x dx
(sec^(n-2) *x *tanx)/(n-1) + (n-2)/(n-1) * integral of sec^(n-2) * x dx, n doesn’t equal 1
Integral of tanxdx
-ln|cosx| + C = ln|secx| + c
Integral of cotxdx
ln|sinx| + c
Integral of secxdx
ln|secx + tanx| + C
Integral of cscxdx
-ln|cscx + cotx| + C
