Integrals and anti-derivatives topics
- Riemann Sums
- Definite integrals
- Fundamental Theorem of Calculus
A Riemann Sum equivalent to the definite integral from a to b of f(x)dx
limit as delta x -> 0 of the sum of f(xi)deltax for a
a Riemann sum is an approximation:
Given the anti derivative of any given function, what will you almost always need to add at the end?
C, the constant that will go to 0 once differentiated
integral from a to b f(x)dx
f(b) - f(a) where F' = f
Derivative with respect to x of an integral from a to x? ( f(t) and dt )
f(x)
d/dx of a to g(x) f(t)dt?
f(g(x))*g'(x)
Given the fact that an antiderivative is the opposite direction of the norm, what must we consider?
We do not have the same level of certainty: the constant, C, accounts for this.
SS signifys antiderivative: SS sec u * tan u du
sec u + C
SS csc u * cot u du
-csc u + C
SS tan u du
ln |sec u| +C
What is the antiderivative of cot(x)?
1. cot(x)=cos(x)/sin(x)
2. Let t=sin(x);
=> dt=cos(x)dx.
3. Integral 1/t == ln |t|
=> Integral cot(x) = ln | sin(x) | + C
ln(|x|)+C
Integral(square(csc(x)))
-cot(x) + C
SS secu du
ln| secu + tanu | + C
SS cscu du
-ln| csc u + cot u | +c
SS du(sqrt (1-u^{2) })
arcsin u + C
SS du/(1+u^{2})
arctanu + C
SS du/(u*sqrt(u^{2} -1))
arcsec u u du
SS e^{u} du
e^{u} + C
SS a^{u} du
a^{u}/ln a
SS du/u
ln |u| + C
Displacement
Integral of velocity with respect to time
Distance
Absolute value of the integral of velocity with respect to time:
distance is different from displacement. Cannot have negative distance.
Area between f and g, where f(x) >= g(x) from a to b
SS from a to b of ( f(x) - g(x) ) dx
For any problem dealing with change:
If R(t) represents a rate of change, then the SS fomr a to b R(t)dt represents...
the accumulated amount of change during the timer interval from a to b.
MPH ==> total miles driven (minus negative displacement)
10 gals/sec ==> how many gallons are in the pool
etc etc
Volume by disks
1. Rotate function about x-axis- dx rectangles become dx-height cylinders e.g "disks"
2. Same approximating model- limit as dx goes to 0 of sum of disks
3. Volume of each disk ==> pi*r^{2}*h = pi * ( f(x) )^{2} * dx
4. SS pi*f(x)^{2} dx
Volume by washers
dV = pi(R^{2} - r^{2}) dx
How do you compute the Left Riemann Sum of a function?
For the left Riemann sum, approximating the function by its value at the left-end point gives multiple rectangles with base Δx and height f(a + iΔx). Doing this for i = 0, 1, ..., n − 1, and adding up the resulting areas gives:
The left Riemann sum amounts to an overestimation if f is ?
The left Riemann sum amounts to an overestimation if f is monotonically decreasing on this interval, and an underestimation if it is monotonically increasing.
What is the Left Riemann Sum of a function?
- Let
- If for all i, then S is called a left Riemann sum.
Draw a diagram of the Left Riemann sum of x^{3}over [0,2] using 4 subdivisions
Draw a diagram of the right Riemann sum methods of x^{3}over [0,2] using 4 subdivisions
Riemann sum methods of x^{3}over [0,2] using middle subdivisions
USE MRAM
Draw a diagram of Riemann sum methods of x^{3}over [0,2] using trapezoidal subdivisions.
Riemann sum methods of x3over [0,2] using trapezoidal subdivisions
Compute the area under the curve of y = x^{2} between 0 and 2 using Riemann's method.
- The interval [0, 2] is firstly divided into n subintervals, each of which is given a width of ; these are the widths of the Riemann rectangles (hereafter "boxes").
- Because the right Riemann sum is to be used, the sequence of x coordinates for the boxes will be .
- Therefore, the sequence of the heights of the boxes will be .
- It is an important fact that , and .
- The area of each box will be and therefore the nth right Riemann sum will be:
- Take the limit as n → ∞
This method agrees with the definite integral as calculated in more mechanical ways:
Draw the diagrams for the 4 Riemann Sum methods for an arbitrary function f.
Four of the Riemann summation methods for approximating the area under curves.
- Right and leftmethods make the approximation using the right and left endpoints of each subinterval, respectively.
- Maximum and minimum methods make the approximation using the largest and smallest endpoint values of each subinterval, respectively.
- The values of the sums converge as the subintervals halve from top-left to bottom-right.