## [answered] MATH260Week 6 Lab Name: Antiderivatives According to the fi

• Use own words.
• What are antiderivatives? How are they connected to derivatives?
• How do we determine an antiderivative? What formulas can we use?
• What is an indefinite integral? How is it related to antiderivatives?
• Why does the indefinite integral require +c on the end of its solutions? Why is the +c not needed for a definite integral?
• Why is (x+5)3 a correct anitiderivative of 3(x+5)2, whereas (2x+5)3 is not a correct antiderivative of 3(2x+5)2 ?
• What is the power rule for integration? How do we use it?
• What is a definite integral? How is it connected to area under the curve?
• How do we find area under the curve using Riemann Sums?
• What is the Fundamental Theorem of Calculus? How do we use it to evaluate a definite integral?
• How?do?we?solve?an?integral?using?substitution?

Use 1 or 2 examples.

MATH260?Week 6 Lab Name: Antiderivatives

According to the first part of the fundamental theorem of calculus, the antiderivative reverses the

derivative. If f(x) is a derivative, F(x) is the antiderivative.

Directions: Look at the examples below and answer questions 1 and 2.

Let f(x) be a derivative and F(x) be the anitderivative.

a.) f(x) = 3x2

2 +1

3x

F( x )=

=x 3

2+ 1 b.) f(x) = 5x ? 6 3 c.)

1 +1 F( x )= 0+1 5x

6x

?

1+ 1

1 5

= x 2?6 x

2 f (x)= ? x? 1 3

+ ?4 +1 1

x4 4 x3 3 x

3

1

F( x )=

?

= x3+ 3

4

?3 4

3x

3 2) Find the antiderivative of f(x) = 3x5 + x ? 4. Show all work. 3) How is an antiderivative related to a derivative? How can that relationship help you to check

your antiderivatives and integrals answers? 4) What do the derivatives of the following antiderivatives have in common?

F(x) = 3x2 + 5 , G(x) = 3x2 + 9 , H(x) = 3x2 ? 11 Indefinite Integrals: ? f (x) dx=F (x)+C Used for finding the general form of the antiderivative. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Constant of Integration: C is the constant of integration that must be attached to an indefinite integral.

It represents any of the constants that could be part of F(x).

Directions: Look at the examples, then find each of the integrals below. ? 4 x3 ?6 x dx ? 3 x+ 5 dx

3 2

x +5 x +C

F(x) = 2 F(x) = 2 3 3 2

2 x 2 ? +C

x F(x) = 6) Find the integral for f (x)= ? x + 2 4 x?5? dx

?

?? ? 3 ? x+ x 2 dx

F(x) = 4 x ?3 x + C 4

+7 x?1

2

. Show all work.

x 4 x?5? 3+ C

1

?

12 Integration by Substitution

If part of the function is a derivative of the other part of the function, then integration by substitution can

be applied. If the form is true, the following method will apply.

1)

2)

3)

4) Label part of the function u and then find du, the derivative of u.

Substitute u and du into the integral to replace all expressions in x.

Integrate the substituted function.

Replace u with the original expression. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Examine each example below and answer questions 7 and 8.

Integral: Let u = 2 ? 2 x ( x +4 ) dx

2 Integral: 2 x + 4 , then du = 2x dx x3

? 4 5 dx

( 5 x +9 )

4 Let u = 5 x +9 , then du = 1

du=x3 dx

20 and

Substitution: ? u2 du Substitution: x3

1

? 4 5 dx= 20 ? du

u5

( 5 x +9 ) Solution: and 1

F ( u )= u 3

3 3

1

F ( x )= ( x 2 +4 ) +C

3 7) How is u determined? How is du determined? 8) Why is there a 1/20 in the substitution for ? Solution: and F ( x )= F ( u )= ?1

4 3 20 x dx 4 80 ( 5 x + 9 ) 1 ?1

[

20 4 u4 +C 9) Identify u, du, the substituted integral, the correct integration, and the final answer.

5 ? 3 x 2 ( x 3?2 ) dx Final Answer = 10) Find 3 x 4+ 3 ?9 dx

4 x3 ?

. Show u and du, the substituted integral, and the final answer.

?? 11) Find ? x4

dx

3

. Show u and du, the substituted integral, and the final answer.

( 7 x 5+1 ) Finding C: You have already seen that a general solution to an integral can be found by performing

antidifferentiation. However, because the derivative of a constant is zero, any integral has many

solutions, each differing from the others by C, a constant of integration.

In many applications of integration, you are given enough information to determine a particular solution

(that is, you can find C). To do this, all you need is an ordered pair from the original function. 12) If s?(t) = v(t), and s??(t) = a(t), use the fact that integration reverses differentiation (+ C) and the

given information about s(t) to find the particular solution to the following problem.

If a model rocket?s velocity is given by a(t)= -32 m/s 2 and v(0)= 10 and position at 1 second is

1,000 m, find s(t). Hint: Integrate a(t) to find v(t), then integrate v(t) to find s(t), finding each C using

the given info. Part II: Finding the Area Under a Curve

The area under a curve can be found by filling the area with rectangles, then finding the area of each

rectangle, then adding the areas together. To find an approximate area under a curve, find the total

area for both left and right rectangles, then average the two. These are called Riemann sums and are

the basis of the integral.

--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 13) For the function f(x) = (x ? 2)2 + 2 estimate the area under the curve on the interval [0, 4] using

5 left rectangles, then 5 right rectangles, then find the average of the two areas. Finally, write

and find the exact area by using a definite integral. Give answers in exact form (no decimals).

Sketch five left rectangles: Based on your sketch, is this estimate an over- or underestimate? Why? Fill in the blanks below. What is the width of each rectangle? Width = x1 = f(x1) = x2 = 4

=.8

5 x3 = 4 4 8

+ = =1.6

5 5 5 f(x3) = x4 = 8 4 12

+ = =2.4

5 5 5 f(x4) = x5 = 12 4 16

+ = =3.2

5 5 5 f(x5) = Total area of the left rectangles = f(x2) = f ( 4 )= 86 =3.44

5 25 f ( 8 )= 54 =2.16

5 25 f ( 12 )= 54 =2.16

5

25 f ( 16 )= 86 =3.44

5

25 4

86 54 54 86 344

6+ + + +

=

=13.76

5

25 25 25 25

25 ( ) Sketch five right rectangles: 4 Based on your sketch, is this estimate an over- or underestimate? Why? 4

5 x1 = 4

=.8

5 f(x1) = x2 = 4 4 8

+ = =1.6

5 5 5 f(x2) = x3 = 8 4 12

+ = =2.4

5 5 5 f(x3) = x4 = 12 4 16

+ = =3.2

5 5 5 f(x4) = x5 = 16 4 20

+ = =4

5 5 5 f(x5) = Total area of the left rectangles =

Average of the left &amp; right rectangles = f ( 4 )= 86 =3.44

5 25 f ( 8 )= 54 =2.16

5 25 f ( 8 )= 54 =2.16

5 25 f ( 16 )= 86 =3.44

5

25 f ( 5 ) =11 4 86 54 54 86

324

+ + + +5 =

=12.96

5 25 25 25 25

25 ( ) 669

=13.38

25 The Definite Integral and Area Under a Curve

If you use the above method and let the number of rectangles be infinite (the limit), the area can be

found to a very accurate measure.

To find the sum of the area of an infinite number of rectangles, use the second part of the fundamental

theorem of calculus.

b ? f ( x)dx=F (b)?F (a)

a 14) Using the fundamental theorem of calculus above, write the correct integral and find the exact

area for problem 13 above. Some Basic Rules of the Definite Integral: all f(x) must exist and be continuous on [a, b].

a 1) ? f ( x) dx

a =0 (if the width is zero, there can be no area) b 2) ? f ( x) dx

a a b 3) ? f ( x) dx = - (if you switch the limits of integration, switch the sign) b c ? f ( x) dx =

a b ? f (x)dx

a + ? f (x) dx

c , for a &lt; c &lt; b (If c is between a and b, then you can break the integral in two at c and add them together

to get the total area; i.e., the area from 2 to 5 is equal to the area from 2 to 3 plus 3 to 5.)

b 4) ? kf ( x)dx

a b = k ? f ( x) dx

a (you can move any constant factor of the function outside of the integral, then integrate and

multiply it back over the answer.) b 5) ? [f (x) ? g (x)]dx

a b = b ? f (x) dx ?? g( x ) dx

a a (the integral of a sum or difference is the sum or difference of the integrals) Note: there is no such rule for multiplication and division. 15) Using the rules for definite integrals above, answer the following questions a)?d).

0 a) Why can?t you find the value of this definite integral? ? ? x?1 dx

?2 3 b) Why can?t you find the value of this definite integral? ? x 2+ 3 x +1 dx

3 3 c) Will the following help to find the value of this definite integral?

3 3 ? x (x+ 3)dx=? x

2 1 1 ? x 2 ?( x+ 3) dx

1 3

2 dx ? ? ( x+3) dx

1 Why or why not? d) How is it possible to get a negative area? 16) Find the area under the curve 3 f ( x )=x ?9 x from [-3, 3] by using the fundamental rule of b calculus: ? f ( x)dx=F (b)?F (a)

a . Show all work. 17) Below is the graph of f(x) = x3 9x with the area between the graph and the x-axis from -3 to 3

shaded. How does it help to explain your answer to 16) above? How would you find the actual

area? 18) The derivative gives an expression for the slope of a curve and the definite integral gives the

area under the curve, two very different ideas.

Name two things the derivative and the integral have in common.

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