<Text-field style="_cstyle4" layout="_pstyle3">Exploring the Fundamental Theorem of Calculus</Text-field>

<Text-field style="_cstyle5" layout="_pstyle4">The idea behind Part 1 of the FTOC</Text-field>

Suppose I take a continuous positive function f, and define another function g that gives the area under f from 0 to whatever x-value I'm interested in.

f:=x->(x/3)^3-x+3;

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plot(f(x),x=0..6,y=0..5,title=`y = f(x)`);

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Then g(1) is the area under f(x) for x in [0, 1], g(2) is the area under f(x) for x in [0, 2], g(3) is the area under f(x) for x in [0, 3], etc. We can calculate some values of g(x) via integrals.

g(1)=Int(f(x),x=0..1); plot(f(t),t=0..1,title="area under f from 0 to 1"); g(1)=evalf(int(f(x),x=0..1));

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g(2)=Int(f(x),x=0..2); plot(f(t),t=0..2,title="area under f from 0 to 2"); g(2)=evalf(int(f(x),x=0..2));

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

g(3)=Int(f(x),x=0..3); plot(f(t),t=0..3,title="area under f from 0 to 3"); g(3)=evalf(int(f(x),x=0..3));

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Do the values above look reasonable? Check the area under the graphs of f(x). For example, to see whether you believe that g(3) is indeed 5.25, estimate the area under f(x) over [0,3].

So really, what I'm doing is saying that g(b) is the area under f(x) from 0 to b.

g(b)=Int(`f(x)`,x=0..b);

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We'll want to use lots of different values for b, so we might as well make our new function g a function of the variable x. Since it doesn't matter what variable I use in the integral, I'm going to call the integrand f(t).

g(x)=Int(`f(t)`,t=0..x);

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With the particular function I'm using, I can define g(x) the way Maple defines functions. Then I can evaluate the function the "easy" way.

g:=x->int(f(t),t=0..x);

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g(1.); g(2.); g(3.);

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This provides evidence that the way I defined g(x) gives a perfectly valid function.

For instance, I can plot g(x):

plot(g(x),x=0..5,title=`y=g(x)`);

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<Text-field style="_cstyle5" layout="_pstyle4">Integrals and Derivatives</Text-field>

We can define a function to be not only the area under another function, but the integral of another function. For instance, suppose this time we say that g(x) is the integral of a continuous function f(t) from 0 to "x".

restart;

g(x)=Int(f(t),t=0..x);

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

Since g(x) is a function, we can do function-type things with it. We already saw that we can graph LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2OVEhRicvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GJy8lJXNpemVHUSMxMkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSp1bmRlcmxpbmVHRjcvJSpzdWJzY3JpcHRHRjcvJSxzdXBlcnNjcmlwdEdGNy8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStiYWNrZ3JvdW5kR0ZDLyUnb3BhcXVlR0Y3LyUrZXhlY3V0YWJsZUdGNy8lKXJlYWRvbmx5R0Y3LyUpY29tcG9zZWRHRjcvJSpjb252ZXJ0ZWRHRjcvJStpbXNlbGVjdGVkR0Y3LyUscGxhY2Vob2xkZXJHRjcvJTBmb250X3N0eWxlX25hbWVHUSlfY3N0eWxlM0YnLyUqbWF0aGNvbG9yR0ZDLyUvbWF0aGJhY2tncm91bmRHRkMvJStmb250ZmFtaWx5R0YxLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy8lKW1hdGhzaXplR0Y0LUYjNiUtRiw2OVEiZ0YnRi9GMkY1RjhGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW5GZ25Gam4tSSNtb0dGJDYzUTAmQXBwbHlGdW5jdGlvbjtGJy8lJWZvcm1HUSZpbmZpeEYnLyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjcvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdGXnAvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKG1heHNpemVHUSlpbmZpbml0eUYnLyUobWluc2l6ZUdRIjFGJy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUwZm9udF9zdHlsZV9uYW1lR0ZWLyUlc2l6ZUdGNC8lK2ZvcmVncm91bmRHRkMvJStiYWNrZ3JvdW5kR0ZDLUkobWZlbmNlZEdGJDYjLUYjNiMtRiw2OVEieEYnRi9GMkY1RjhGO0Y9Rj9GQUZERkZGSEZKRkxGTkZQRlJGVEZXRllGZW5GZ25Gam5GKw==; we can also differentiate g. We'll use a different f(t) this time, to illustrate that f(t) doesn't have to be positive.

f:=t->t^2-5*t;

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtaUdGJTY5USJmRigvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GKC8lJXNpemVHUSMxMkYoLyUlYm9sZEdRJmZhbHNlRigvJSdpdGFsaWNHUSV0cnVlRigvJSp1bmRlcmxpbmVHRjgvJSpzdWJzY3JpcHRHRjgvJSxzdXBlcnNjcmlwdEdGOC8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GKC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRigvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjsvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYoLyUqbWF0aGNvbG9yR0ZELyUvbWF0aGJhY2tncm91bmRHRkcvJStmb250ZmFtaWx5R0YyLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGKC8lKW1hdGhzaXplR0Y1LUkjbW9HRiU2M1EjOj1GKC8lJWZvcm1HUSZpbmZpeEYoLyUmZmVuY2VHRjgvJSpzZXBhcmF0b3JHRjgvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYoLyUncnNwYWNlR0ZbcC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobWF4c2l6ZUdRKWluZmluaXR5RigvJShtaW5zaXplR1EiMUYoLyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJTBmb250X3N0eWxlX25hbWVHRlgvJSVzaXplR0Y1LyUrZm9yZWdyb3VuZEdGRC8lK2JhY2tncm91bmRHRkctRiQ2JS1GLTY5USJ0RihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZKRkxGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG8tRl9vNjNRJyZyYXJyO0YoL0Zjb1EhRihGZW9GZ28vRmpvUSQwZW1GKC9GXXBGYXJGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtRiQ2JS1GJDYjLUklbXN1cEdGJTYlRmhxLUkjbW5HRiU2OVEiMkYoRjBGM0Y2L0Y6RjhGPEY+RkBGQkZFRkhGSkZMRk5GUEZSRlRGVkZZRmVuRmduL0ZqblEnbm9ybWFsRihGXG8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYoLUZfbzYzUSgmbWludXM7RihGYm9GZW9GZ28vRmpvUTBtZWRpdW1tYXRoc3BhY2VGKC9GXXBGaHNGXnBGYHBGYnBGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHEtRiQ2JS1GW3M2OVEiNUYoRjBGM0Y2Rl5zRjxGPkZARkJGRUZIRkpGTEZORlBGUkZURlZGWUZlbkZnbkZfc0Zcby1GX282M1ExJkludmlzaWJsZVRpbWVzO0YoRmJvRmVvRmdvRmByRmJyRl5wRmBwRmJwRmVwRmhwRmpwRlxxRl5xRmBxRmJxRmRxRmhxNyMtX0YpSSxtcHJpbnRzbGFzaEdGKDYkNyM+SSJmR0YoZio2I0kidEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLCYqJClGXHUiIiMiIiJGZHUqJiIiJkZkdUZcdUZkdSEiIkYoRihGKDcjRmp0

plot(f(t),t=0..10);

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

g:=x->int(f(t),t=0..x);

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

What should g(x) look like? At first, f(t) is negative. So 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 should be getting more and more negative as x goes from 0 to about 5. Then 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 starts adding in positive area. Once x is about 8, the positive and the negative areas should cancel out, and from that point on 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 will be positive and increasing.

plot(g(x),x=0..10);

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

What else? Well, let's check the derivative of g(x).

diff(g(x),x);

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

This should look familiar!

f(t);

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtRiQ2Iy1JJW1zdXBHRiU2JS1JI21pR0YlNjlRInRGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dRJXRydWVGKC8lKnVuZGVybGluZUdGPS8lKnN1YnNjcmlwdEdGPS8lLHN1cGVyc2NyaXB0R0Y9LyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYoLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GKC8lJ29wYXF1ZUdGPS8lK2V4ZWN1dGFibGVHRj0vJSlyZWFkb25seUdGQC8lKWNvbXBvc2VkR0Y9LyUqY29udmVydGVkR0Y9LyUraW1zZWxlY3RlZEdGPS8lLHBsYWNlaG9sZGVyR0Y9LyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RigvJSptYXRoY29sb3JHRkkvJS9tYXRoYmFja2dyb3VuZEdGTC8lK2ZvbnRmYW1pbHlHRjcvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YoLyUpbWF0aHNpemVHRjotSSNtbkdGJTY5USIyRihGNUY4RjsvRj9GPUZBRkNGRUZHRkpGTUZPRlFGU0ZVRldGWUZlbkZobkZqbkZcby9GX29RJ25vcm1hbEYoRmFvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGKC1JI21vR0YlNjNRKCZtaW51cztGKC8lJWZvcm1HUSZpbmZpeEYoLyUmZmVuY2VHRj0vJSpzZXBhcmF0b3JHRj0vJSdsc3BhY2VHUTBtZWRpdW1tYXRoc3BhY2VGKC8lJ3JzcGFjZUdGanAvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKG1heHNpemVHUSlpbmZpbml0eUYoLyUobWluc2l6ZUdRIjFGKC8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUwZm9udF9zdHlsZV9uYW1lR0Znbi8lJXNpemVHRjovJStmb3JlZ3JvdW5kR0ZJLyUrYmFja2dyb3VuZEdGTC1GJDYlLUZkbzY5USI1RihGNUY4RjtGZ29GQUZDRkVGR0ZKRk1GT0ZRRlNGVUZXRllGZW5GaG5Gam5GXG9GaG9GYW8tRl5wNjNRMSZJbnZpc2libGVUaW1lcztGKEZhcEZkcEZmcC9GaXBRJDBlbUYoL0ZccUZec0ZdcUZfcUZhcUZkcUZncUZpcUZbckZdckZfckZhckZjckYxNyMsJiokKUkidEdGKCIiIyIiIkZmcyomIiImRmZzRmRzRmZzISIi

Is this a coincidence?

<Text-field style="_cstyle5" layout="_pstyle4">Antiderivatives -- the idea behind Part 2 of the FTOC</Text-field>

Two of your Lab 7 Homework questions will ask you to work through the ideas illustrated in this section.

Consider the integral 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 . Enter the integrand as a function:

f:=x->x*sin(x^2)+3*x^2;

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

Guess an antiderivative ("F")

F:=x->-1/2*cos(x^2)+x^3; (This guess might require some trial and error!)

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

Differentiate and check that your F(x) is really an antiderivative of f(x).

diff(F(x),x);

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

Evaluate F(x) at the limits of integration:

F(3)-F(-1); evalf(%);

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

Check this result by having Maple evaluate the original integral:

Int(x*sin(x^2)+3*x^2,x=-1..3); value(%); evalf(%);

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNictSShtc3Vic3VwR0YlNictSSNtb0dGJTY0USYmaW50O0YoLyUlZm9ybUdRIUYoLyUmZmVuY2VHUSZmYWxzZUYoLyUqc2VwYXJhdG9yR0Y4LyUnbHNwYWNlR1EkMGVtRigvJSdyc3BhY2VHRj0vJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKG1heHNpemVHUSlpbmZpbml0eUYoLyUobWluc2l6ZUdRIjFGKC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RigvJSVzaXplR1EjMTJGKC8lK2ZvcmVncm91bmRHUS5bMTQ0LDE0NCwxNDRdRigvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XUYoL0krbXNlbWFudGljc0dGJVEmaW5lcnRGKC1JI21uR0YlNjlRKSZtaW51czsxRigvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GKC8lJXNpemVHRlUvJSVib2xkR0Y4LyUnaXRhbGljR0Y4LyUqdW5kZXJsaW5lR0Y4LyUqc3Vic2NyaXB0R0Y4LyUsc3VwZXJzY3JpcHRHRjgvJStmb3JlZ3JvdW5kR1EqWzAsMCwyNTVdRigvJStiYWNrZ3JvdW5kR0Zlbi8lJ29wYXF1ZUdGOC8lK2V4ZWN1dGFibGVHRjgvJSlyZWFkb25seUdRJXRydWVGKC8lKWNvbXBvc2VkR0Y4LyUqY29udmVydGVkR0Y4LyUraW1zZWxlY3RlZEdGOC8lLHBsYWNlaG9sZGVyR0Y4LyUwZm9udF9zdHlsZV9uYW1lR0ZSLyUqbWF0aGNvbG9yR0ZecC8lL21hdGhiYWNrZ3JvdW5kR0Zlbi8lK2ZvbnRmYW1pbHlHRl9vLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGKC8lKW1hdGhzaXplR0ZVLUZqbjY5USIzRihGXW9GYG9GYm9GZG9GZm9GaG9Gam9GXHBGX3BGYXBGY3BGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHFGZnFGaHFGW3IvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMkYoLyUvc3Vic2NyaXB0c2hpZnRHUSIwRigtRiQ2JS1GJDYlLUYkNiUtSSNtaUdGJTY5USRzaW5GKEZdb0Zgb0Zib0Zkb0Zmb0Zob0Zqb0ZccEZfcEZhcEZjcEZlcEZocEZqcEZccUZecUZgcUZicUZkcUZmcUZocUZbci1GMDYzUTAmQXBwbHlGdW5jdGlvbjtGKC9GNFEmaW5maXhGKEY2RjlGO0Y+RkBGQkZERkdGSkZMRk5GUEZTL0ZXRl5wRlktRiQ2JS1GMDYzUSIoRigvRjRRJ3ByZWZpeEYoL0Y3RmdwRjkvRjxRLnRoaW5tYXRoc3BhY2VGKC9GP0ZfdC9GQUZncEZCRkRGR0ZKRkxGTkZQRlNGZXNGWS1GJDYjLUYkNiMtSSVtc3VwR0YlNiUtRl1zNjlRInhGKEZdb0Zgb0Ziby9GZW9GZ3BGZm9GaG9Gam9GXHBGX3BGYXBGY3BGZXBGaHBGanBGXHFGXnFGYHFGYnFGZHFGZnEvRmlxUSdpdGFsaWNGKEZbci1Gam42OVEiMkYoRl1vRmBvRmJvRmRvRmZvRmhvRmpvRlxwRl9wRmFwRmNwRmVwRmhwRmpwRlxxRl5xRmBxRmJxRmRxRmZxRmhxRltyL0ZhckZlci1GMDYzUSIpRigvRjRRKHBvc3RmaXhGKEZddEY5Rl50L0Y/UTJ2ZXJ5dGhpbm1hdGhzcGFjZUYoRmF0RkJGREZHRkpGTEZORlBGU0Zlc0ZZLUYwNjNRMSZJbnZpc2libGVUaW1lcztGKEZjc0Y2RjlGO0Y+RkBGQkZERkdGSkZMRk5GUEZTRmVzRllGaXQtRjA2M1EiK0YoRmNzRjZGOS9GPFEwbWVkaXVtbWF0aHNwYWNlRigvRj9GYXZGQEZCRkRGR0ZKRkxGTkZQRlNGZXNGWS1GJDYlRl1yRmp1RmZ0LUknbXNwYWNlR0YlNiYvJSdoZWlnaHRHUScwLjB+ZXhGKC8lJndpZHRoR1EnMC4zfmVtRigvJSZkZXB0aEdGanYvJSpsaW5lYnJlYWtHUSVhdXRvRigtRjA2NFEwJkRpZmZlcmVudGlhbEQ7RihGW3RGNkY5RjtGPkZARkJGREZHRkpGTEZORlBGU0ZWRllGZm5GaXQ3Iy1JJEludEc2JCUqcHJvdGVjdGVkR0YqNiQsJiomLUkkc2luR0ZpdzYjKiQpSSJ4R0YoIiIjIiIiRmV4RmN4RmV4RmV4KiYiIiRGZXhGYnhGZXhGZXgvRmN4OyEiIkZneA==

First, note that we got the same result by substituting the limits of integration in to the antiderivative and subtracting (F(3)-F(-1)) as we did by just having Maple evaluate the integral. This verifies the FTOC, Part 2, for this particular integral.

Second, we should evaluate whether the answer make sense in terms of the area under the graph of f(x).

plot(f(x),x=-1..3);

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

The numerical value of the integral (28.7) seems to be in the right ball park. The function is positive, and the area under the function to the right of the x-axis is less than that of a triangle of height 30 and base length 3.