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The FTC Part 2, Total Change/Area & U-Sub

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Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty and can hold 6 units, at what time will it overflow? A. log(7)/3 B. (1/3)log(13/7) C. 3 log (13/7) D. 3log(7) E. Never

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Question from Test 1 Liquid drains into a tank at the rate 21e -3t units per minute. If the tank starts empty and can hold 6 units, at what time will it overflow? A. log(7)/3

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The Total Change Theorem The integral of a rate of change is the total change from a to b. (displacement) (still from last weeks notes)

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The Total Change Theorem Ex: Given Find the displacement and total distance traveled from time 1 to time 6. Displacement: (negative area takes away from positive) Total Distance: (all area counted positive)

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Total Area Find the area of the region bounded by the x-axis, y-axis and y = 2 – 2x. First find the bounds by setting 2 – 2x = 0 and by substitution 0 in for x

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Total Area Ex. Find the area of the region bounded by the y-axis and the curve

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Fundamental Theorem of Calculus (Part 1) (Chain Rule) If f is continuous on [a, b], then the function defined by is continuous on [a, b] and differentiable on (a, b) and

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Fundamental Theorem of Calculus (Part 1)

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Fundamental Theorem of Calculus (Part 2) If f is continuous on [a, b], then : Where F is any antiderivative of f. ( ) Helps us to more easily evaluate Definite Integrals in the same way we calculate the Indefinite!

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Example

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We have to find an antiderivative; evaluate at 3; evaluate at 2; subtract the results.

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Example

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This notation means: evaluate the function at 3 and 2, and subtract the results.

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Example

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Don’t need to include “+ C” in our antiderivative, because any antiderivative will work.

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Example the “C’s” will cancel each other out.

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Example

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Alternate notation

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Example

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= –1

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Example = –1= 1

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Example

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Given: Write a similar expression for the continuous function:

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Fundamental Theorem of Calculus (Part 2) If f is continuous on [a, b], then : Where F is any antiderivative of f. ( )

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Evaluate: Multiply out: Use FTC 2 to Evaluate:

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What if instead? It would be tedious to use the same multiplication strategy! There is a better way! We’ll use the chain rule (backwards)

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Chain Rule for Derivatives: Chain Rule backwards for Integration:

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Look for:

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Back to Our Example Let

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Our Example as an Indefinite Integral With AND Without worrying about the bounds for now: Back to x (Indefinite):

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The same substitution holds for the higher power! With Back to x (Indefinite):

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Our Original Example of a Definite Integral: To make the substitution complete for a Definite Integral: We make a change of bounds using: When x = -1, u = 2(-1)+1 = -1 When x = 2, u = 2(2) + 1 = 5 The x-interval [-1,2] is transformed to the u-interval [-1, 5]

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Substitution Rule for Indefinite Integrals If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then Substitution Rule for Definite Integrals If g’(x) is continuous on [a,b] and f is continuous on the range of u = g(x), then

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Using the Chain Rule, we know that: Evaluate:

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Looks almost like cos(x 2 ) 2x, which is the derivative of sin(x 2 ). Using the Chain Rule, we know that:

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Evaluate: We will rearrange the integral to get an exact match: Using the Chain Rule, we know that:

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Evaluate: We will rearrange the integral to get an exact match: Using the Chain Rule, we know that:

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Evaluate: We will rearrange the integral to get an exact match: Using the Chain Rule, we know that:

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Evaluate: We will rearrange the integral to get an exact match: We put in a 2 so the pattern will match. Using the Chain Rule, we know that:

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Evaluate: We will rearrange the integral to get an exact match: We put in a 2 so the pattern will match. So we must also put in a 1/2 to keep the problem the same. Using the Chain Rule, we know that:

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Evaluate: We will rearrange the integral to get an exact match: Using the Chain Rule, we know that:

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Evaluate: We will rearrange the integral to get an exact match: Using the Chain Rule, we know that:

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Check Answer:

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Check: Check Answer:

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Check: From the chain rule Check Answer:

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1)Choose u. Indefinite Integrals by Substitution

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1)Choose u. 2)Calculate du. Indefinite Integrals by Substitution

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1)Choose u. 2)Calculate du. 3)Substitute u. Arrange to have du in your integral also. (All xs and dxs must be replaced!) Indefinite Integrals by Substitution

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1)Choose u. 2)Calculate du. 3)Substitute u. Arrange to have du in your integral also. (All xs and dxs must be replaced!) 4)Solve the new integral. Indefinite Integrals by Substitution

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1)Choose u. 2)Calculate du. 3)Substitute u. Arrange to have du in your integral also. (All xs and dxs must be replaced!) 4)Solve the new integral. 5)Substitute back in to get x again.

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Example A linear substitution: Let u = 3x + 2. Then du = 3dx.

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Choosing u Try to choose u to be an inside function. (Think chain rule.) Try to choose u so that du is in the problem, except for a constant multiple.

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Choosing u For u = 3x + 2 was a good choice because (1)3x + 2 is inside the exponential. (2)The derivative is 3, which is only a constant.

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Practice Let u = du =

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Practice Let u = x 2 + 1 du =

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Practice Let u = x 2 + 1 du = 2x dx

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Practice Let u = x 2 + 1 du = 2x dx

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Practice Let u = x 2 + 1 du = 2x dx Make this a 2x dx and we’re set!

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Practice Let u = x 2 + 1 du = 2x dx

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Practice Let u = x 2 + 1 du = 2x dx

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Practice Let u = x 2 + 1 du = 2x dx

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Practice Let u = x 2 + 1 du = 2x dx

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Practice Let u = du =

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Practice Let u = sin(x) du =

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Practice Let u = sin(x) du = cos(x) dx

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Practice Let u = sin(x) du = cos(x) dx

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Practice Let u = sin(x) du = cos(x) dx

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Practice Let u = sin(x) du = cos(x) dx

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Practice Let u = du = An alternate possibility:

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Practice Let u = cos(x) du = –sin(x) dx An alternate possibility:

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Practice Let u = cos(x) du = –sin(x) dx An alternate possibility:

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Practice Let u = cos(x) du = –sin(x) dx An alternate possibility:

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Practice Let u = cos(x) du = –sin(x) dx An alternate possibility:

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Practice Note:

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Practice Note:

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Practice Note: What’s the difference?

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Practice Note: What’s the difference?

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Practice Note: What’s the difference?

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Practice Note: What’s the difference? This is 1!

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Practice Note: What’s the difference?

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Practice Note: What’s the difference? That is, the difference is a constant.

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In-Class Assignment Integrate using two different methods: 1st by multiplying out and integrating 2nd by u-substitution Do you get the same result? (Don’t just assume or claim you do; multiply out your results to show it!) If you don’t get exactly the same answer, is it a problem? Why or why not?

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