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Name:__________________

Below is the graph of f(x) = x2 + 2. The shaded area is the area of concern. We have bounded the area between f(x)and the x-axis over the interval [0,1].

1.)

a. Let n= 2, so partition the area over the interval [0,1] into 2 equal sections by drawing lines up to

the f(x) graph. Use these lines as the right-hand endpoints of rectangles.

b. Determine x . ( Hint x =n

ab )

c. Use the right endpoints to estimate the upper sum of the area under the curve by filling out the

chart below:

Height of rectangle

f(ci)= f(ix

)

Width of

rectangle x

Area = hw

f(ci)x

Rectangle i=12

2

2

+

i

= 22

12

+

2

1

81

2

12

12

2

1=

+

Rectangle i=22

2

2

+

i

=

Total Area:

d.) The area under the curve is 2.33 or 312

. How far off were you with your guess?

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2.)a. Let n= 8, so partition the area over the interval [0,1] into 8 equal sections by drawing lines up to

the f(x) graph. Use these lines as the right-hand endpoints of rectangles.

b. Determine x . ( Hint x =n

ab )

c. Use the right endpoints to estimate the upper sum of the area under the curve by filling out thechart below:

Height of rectangle

f(ci)= f( ix )

Width of

rectangle x

Area = hw

f(ci) x

Rectangle i=12

8

2

+

i

= 28

12

+

8

1=

+

8

12

8

12

Rectangle i=22

8

2

+

i

=

Rectangle i=3

Rectangle i=4

Rectangle i=5

Rectangle i=6

Rectangle i=7

Rectangle i=8

Total Area:

d.) Represent the process of summing up the 8 rectangles you just calculated in the chart, using

sigma/summation notation for representing.

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3.) Draw a picture of the graph f(x) = x2 + 2 over the interval [0,1] such that the area under the curve is

partitioned into an infinite amount of rectangles bounded by the curve.

4.) Let n not be defined by a real number. In other words, n is just n, we will partition the interval [0,1] into n

pieces. Use sigma/summation notation to represent the area under the curve f(x) = x2 + 2 over the interval[0,1] if you partition the area into n rectangles.

5.) Find the limit as n-> of the summation of the n rectangles as you have found in problem number 4. Inother words find

xcf i

n

ix

=

)(lim

1

(Hint: Use the summation formulas from page 399 of your textbook to get rid of sigma notation)

6.) Find F(x) of f(x) = x2 +2. In other words, integrate f(x), or find the antiderivative of f(x).

Remember the notation: F(x) = + 22x dx.

7.) Find +2

1

2 2x dx by completing parts a c below

a.) Use your answer from 7 and find F(2)

b.) Find F(1)

c.) Use the fundamental Theorem of Calculus to find +2

1

2 2x dx = F(2)- F(1)

8.) Explain the link between limits of Riemann Sums and integration.

9.) Challenge: Find the area bounded by the equation y = x2 +2 ; y = x + 2 ; x=0 ; and x = 1