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**calculus 2** **(Part 2 - Integrals & Vector Calculus)**- Limits
- Differential Calculus (Check out the ANIMATION!)
- Integral Calculus (newly updated - 10/99)
- Vector Calculus (updated - 10/98)
- (c) 1997-2001 Dan Bach and B & L Math Enterprises; all rights reserved. Download for personal use only.
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Integral Calculus(Antiderivatives, areas, applications)- 10/99

AntiderivativesThe derivative f'(x) is used to calculate the slope, or rate of change, of a function f(x). But what is the function f(x) the derivative OF? Or, if you prefer better grammar, describe a function F(x) having F'(x) = f(x). Such a function F(x) is called an

antiderivativeof f(x).

Example:What function has a derivative of x^3 ? We know the deriv goes dowm one

- power, and we might think: (x^4) = 4 x^3. But we want x^3 not 4 x^3.
- How do we avoid the 4? Put a 1/4 at the beginning to wait and pounce on the 4 when we do the derivative. Thus if F(x) = (1/4) x^4 , we get F'(x) = (1/4) 4 x^3 = x^3 = f(x).
- Actually F(x) could be (1/4) x^4 + c for any constant c, and F'(x) would still be x^3.
- The antiderivative F(x) is also called the
integralof f(x).- The usual notation for the antiderivative of f(x) is F(x) =f(x) dx .
- We would writex^3 dx = (1/4) x^4 + c.(Check out my new integral.gif!)

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Integration rules!(10/99)I mean it can be fun, challenging, and useful! I also mean there are rules for what do do to look for an antideriv. We just discovered the

Power rule:x^n dx = (1/(n+1)) x^(n+1) + c

Example:x dx = x^(1/2) dx =(2/3) x^(3/2)+ c

Constant multiple rule: c f(x) dx = cf(x) dx(Pulling a constant outside)

Example:3x dx = 3x dx = 3 (1/2) x^2 + c =(3/2) x^2 + c

Sum rule: ( f(x) + g(x)) dx = f(x) dx +g(x) dx

Example:(3x + 4) dx =3x dx +4 dx =(3/2) x^2 + 4 x + c

Chain rule:(aka "u-substitution")

F '(g(x)) g'(x) dx = F '(u) (du/dx) dx =F '(u) du = F(u) + c.

Example:(3x + 4)^20 dx = (1/3)(3x + 4)^20(3 dx)- = (1/3)
u^20du= (1/3) (1/21) u^21 =(1/63) (3x + 4)^21 + c- Here I used u = g(x) = 3x + 4 , so that du = g'(x) dx = 3 dx.
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Calculus of Areas(10/99)Historically, an important problem was figuring out areas. A long time before Newton, Archimedes (c.287 BC, from Greece) and many Chinese mathematicians, were chopping regions up into little slices to estimate their areas.

We can chop up the area A under the graph of y = f(x) by using a bunch of rectangles. Each rectangle has height f(x) and base Dx, so the area of each "strip" is DA = f(x) Dx.

The area A is about the same as the sum of the areas of all those rectangles;

A :=:DA = f(x) Dx. . . (This is called aRiemann Sum.)Here's a Mathematica->QuickTime->GIF Builder animation showing:

The more rectangleswe use,the betterthe approximation to the true area!

The Fundamental Theorem of Calculus(10/99)Let's assume that F(x) is an antiderivative of f(x), that is, F'(x) = f(x).

The surprising part is that F(x) represents the

areaunder the graph of y = f(x). Can this be true? let's define A(x) to be the area under y = f(x) from 0 to x (let's pretend we're in the first quadrant). Then A'(x) would be dA/dx. But the extra area under y = f(x) from x to x + dx is about the same as the area of the skinny rectangle of height f(x) and base dx, so dA = f(x) dx, and so dA/dx = f(x).Here we have that

the area function is an antiderivativeof f(x); A'(x) = f(x).Wow. So to compute an area, we can just compute an antiderivative of f(x)! (not factorial)

If we want the area A from x = a to x = b, we can subtract area functions;

if F(x) is the area from 0 to x : F(x) =f(x) dx , then

A = F(b) - F(a) .

Other applications of integrals(9/00)

"What are we gonna use this stuff for?"- That's what students always ask (or at least wonder), so here are some examples:

1. Work:The amount of "work" you do pushing an object is equal to (how hard you push) times (how far you push it). ThenWork = Force * Distance .If the force is not constant then we have to add up lots of little bits of (force) * (distance) for short distances along our path.

(Work done) =(force at x) (Dx) --> W= F(x) dx

2. Savings account buildup:If you put money into an account and leave it there, it grows by compound interest, exponentially. But if you deposit a little bit, say each month, then each little bit earns interest for a diffferent amount of time (like my students!). This cries out for an integral, where r is annual int rate and n = # of years...

(Amount in account)= (from t = 0 to 12n)

[ (depos at month t) * (int for n - t mos) ]

-->(from t = 0 to 12n)e^(r(n - t)) dt

Vector Calculus(updated 3/99) . . . (top of page)

**(xyz-coords, 3D curves, surfaces)**Pictures done in Mathematica

The xyz-coordinate system; points, lines, planes(back to vec calc)

- Just like a point (x,y) in the plane, or "2-space," means "x over and y up," we can
- have a point (x,y,z) in "3-space" meaning "x forward, y over, and z up."
- See the pictures and follow the bouncing ball.

Parametric curves in 2D and 3D; arclength, curvature(back to vec calc)

- A curve can be
parametrizedby a variable t (such as time), where its coords (two or three) are functions of t. Let's show one 2D curve and one 3D curve:- Here's a spiral curve in the xy-plane:
x = t cos(t) , y = t sin(t) ; 0 < t < 6 Pi . (click here for picture of spiral)

- Now here's a "helix" curve in 3-space:
x = 3 cos(t) , y = 3 sin(t) , z = t ; 0 < t < 6 Pi . (click here for picture of helix)

- The
arclengthis the distance an ant would crawl along the curve; it can be calculated by an appropriate integral hinging on the Pythagorean Theorem.The

curvatureis how fast the ant has to turn, in, say, degrees per centimeter.

Functions of two variables; surfaces(back to vec calc)

- Think of z as a function of x and y ;
z = f(x,y).- For example we could have z = x^2 - y^2 + 10.
- Then for each point (x,y) in the plane, we put a point (x, y, x^2 - y^2 + 10) in 3-space. One is (3, 2, 15). The collection of all these points forms a
surface, which is thegraphof z = f(x,y). Here's part of this surface:

Level curves and gradient vectors(back to vec calc)A

level curvefor level c of a function z = f(x,y) is the set of points (x,y) in the plane for which f(x,y) = c. These are also calledcontours.Here are some level curves for z = x^2 - y^2:

Double, triple integrals; area, volume, surface area(back to vec calc)In first-year calculus you learned that an

areaunder a curve y = f(x) over an interval [a,b] can be calculated by doing the integral f(x) dx , where the integral goes from a to b.Now in 3D calculus we can calculate a

volumeby doing an integral of z over a certain region in the xy-plane: V = f(x,y) dA , where the (x,y) is over a region R and dA = dx dy. Thisdouble integralis evaluated from the "inside out."The

surface areais more complicated, involving the slope and direction of the surface. There are also triple (and higher!) integrals.

Cylindrical and spherical coordinates(9/98)(back to vec calc)These are extensions of the idea of polar coordinates; see Trigonometry.

In 3-dimensional (x, y, z) space, we can use polar coords (r, t) on the xy-plane, together with the z-coordinate z, to get a point

(r, th, z), called "cylindrical coordinates," where th is a "theta" and we have the relations

x = r cos th , y = r sin th , z = z(cylindrical to rectangular) . . . and . . .

r^2 = x^2 + y^2 , tan th = y/x , z = z(rectangular to cylindrical)They're called cylindrical because the graph of

r = cis a cylinder of radius c , because th and z aren't mentioned, so they can be anything; around and around or up and down, but always c units away from the z-axis. (picture when I get to it . . . if ever.)

- An alternative to polar coords is to use the distance
p(rho, not "p") in space from the origin O to the point P, the angleth(theta) from the x-axis to the projection of the ray OP onto the xy-plane, and the angleø(phi) from the z-axis down to the ray OP. The point(p, th, ø)is said to be in "spherical coordinates," because the graph ofp = cis a sphere of radius c, the longitude th and the latitude ø being free to range. (picture next in line.)(top of page)

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