Calculus 2 Review ( Test 1)

• Formal Formula to finding the area between curves

  A = $lim\ n\ \to\ \infty\ \sum_{i=1}^{n}\left[\ f\left(xi\right)\ -\ g\left(xi\right)\right]△x$
• Informal Formula to find area between curves

  A = $\int_{a}^{b}\left[f\left(x\right)-g\left(x\right)\right]dx$
• Steps to finding the area between curves

  1st. Draw a picture (if needed) of the graphs

  2nd. Decide which function goes first ( Top graph - Bottom graph)

  3rd. Simplify the equation and integrate ( find antiderivative )

  4th. Evaluate the equation at the given x values
• Formula of finding the area between the curves respect to y

  $\int_{c}^{d}\left[f\left(y\right)-g\left(y\right)\right]dy$
• Steps to finding the area between curves (for y )

  1st. Draw a picture of the graphs

  2nd. Find the points of intersection, f(y) = g(y)

  3rd. Decide which function goes first ( Right graph - Left graph )

  4th. Simplify the equation and integrate ( find antiderivative )

  5th. Evaluate the equation at the given y-values
• Volume Formula of Rotation/Revolution

  V= pi * integral from a to b ( R(x) )^2 * △X
• The Disk Method

  Horizontal Axis of Revolution:

  V = $\pi\int_{a}^{b}\left[\ R\left(x\right)\right]^{2}dx$

  Vertical Axis of Revolution:

  V = $\pi\int_{c}^{d}\left[\ R\left(y\right)\right]^{2}dy$
• Steps in finding the volume of a solid formed by revolving the region bounded by a graph and the x-axis (Disk Method)

  1st: Draw out the graph and find the intersection points, if not already given

  2nd: Write out the formula using the information given

  3rd: The R(x) value is the 1st function given, the x-axis would be x=0

  4th: Integrate the volume formula and solve
• Steps in finding the volume of a solid formed by revolving the region bounded by two graphs

  1st: Find the point of intersection, the values you find are your upper and lower bounds ( a and b )

  2nd: Draw the graph ( getting a visual )

  3rd: Set up the formula, making sure you follow the rules ( upper graph - lower graph ) for the R(x) value

  4th: Integrate and solve for the volume
• The Washer Method

  Finding the volume of the area between two graphs with the information of the values between the graphs and the x-axis, and the whole distance, using the value to subtract the two graphs.

  V = $\pi\int_{a}^{b}\left(\left[\ R\left(x\right)\right]^{2}-\left[r\left(x\right)\right]^{2}\right)dx$
• Steps to finding the volume of the solid formed by revolving the graph using the Washer Method

  1st: Find the intersection points

  2nd: Graph the functions and remember ( top graph - bottom graph )

  3rd: Integrate and solve the equation

  * Remember that there is a hole between the volume and the axis when it is revolved around the named axis ( x or y axis )
• Volume of Cylindrical Shell (Horizontal Axis of Rotation)

  V= $2\pi\int_{c}^{d}p\left(y\right)h\left(y\right)dy$
• Volume of Cylindrical Shell (Vertical Axis of Rotation x= number)

  V= $2\pi\int_{a}^{b}p\left(x\right)h\left(x\right)dx$
• Volume of Cylindrical Shell

  p(x) or p(y) = usually x or y if the axis is x-axis or y-axis

  h(x) or h(y) = usually the equation given
• How to find the volume of a Cylindrical Shell on another axis ( x= )

  If the line is x= a number and the equation is on the right of the x line, then it will be x - (- number)

  If the line is on the left side, it will be ( number - x )
• How to find the volume of a Cylindrical Shell on another axis ( y=)

  If the line is y= a number and the equation is above the y line, then it will be number - (equation or y )

  If the line is below the equation, then it will be (equation) - a number
• Average value of a function formula

  $\frac{1}{b-a}\int_{a}^{b}f\left(x\right)dx$
• Mean Value Theorem for Integrals

  If f is continuous on [a,b], then there exists a number c that belongs to [a,b] such that,
• Mean Value Theorem for IntegralsIf f is continuous on [a,b], then there exists a number c that belongs to [a,b] such that,

  f(c) = f(ave)

  f(c) = $\frac{1}{b-a}\int_{a}^{b}f\left(x\right)dx$

  So, $\int_{a}^{b}f\left(x\right)dx =$$(b-a)f(c)$
• Force = mass(acceleration)

  F= m(a)

  F= m(d^2s/dt^2)
• Position, Velocity, Acceleration

  s(t) = s'(t) = s''(t)

  s(t)= v(t) = a(t)
• If acceleration is constant, then the Force is also constant

  Work done is also constant
• W= FD

  F= measured in Newtons

  D= measured in meters

  W= measured in N=M or Joules
• Work formula as a limit/integral

  W= $\int_{a}^{b}f\left(x\right)dx$
• Hooke's Law (Springs)

  F= kd

  k= spring constant

  d= the distance the spring is compressed or stretched
• Newton's Law of Universal Gravitation

  F = G [(m1*m2)/r^2]
• Columb's Law

  F= k[(q1*q1)/r^2]

  q1 = charge 1

  q2 = charge 2

  r = distance between charges
• When using the disk/washer method and the graph(s) are being rotated about the y-axis, integrate in terms of y
• When using the disk/washer method and the graph(s) are being rotated about the x-axis, integrate in terms of x