# Thread: 6 problems! Any help appreciated :)

1. ## 6 problems! Any help appreciated :)

1. Use midpoints to approximate the area under the curve (see link) on the interval [0,1] using 10 equal subdivisions.

http://imagizer.imageshack.us/v2/800x600q90/707/5b9m.jpg

3.157---my answer (but I don't understand midpoints)
3.196
3.407
2.078
2.780

2. Use right-hand endpoints and 6 equal subdivisions to approximate the area beneath the curve on the interval [0, 6].

http://imagizer.imageshack.us/v2/800x600q90/38/7ruq.jpg

0.9243
1.405
1.897
1.682

3. The table below gives data points for the continuous function y = f(x)

http://imagizer.imageshack.us/v2/800x600q90/706/khd9.jpg

Approximate the area under the curve y = f(x) on the interval [0, 2] using left-hand endpoints and 10 equal subdivisions. You get Area ≈

96.8
88.8
90.8
444

4. Consider the curve and the region under f (x) between x = 1 and x = 3, which is graphed below.

http://imagizer.imageshack.us/v2/800x600q90/23/f1o5.jpg

Suppose L is the left-hand endpoint Riemann sum with 15 subdivisions, R is the right-hand endpoint Riemann sum with 15 subdivisions, and A is the true area of this region. Which of the following is correct?

R < L < A
L < A < R----my answer
L = A = R
R < A < L
A < R < L

5. The function y = f(x) is graphed below:

http://imagizer.imageshack.us/v2/800x600q90/841/a45g.jpg

Which of the following Riemann sums yields the exact area under the curve on the interval [0, 6]?

I. R=E(above=4)below=k=1 f(wk)deltaxk, where subdivisions are at {0, 2, 3, 4, 6} and right-hand endpoints are used.

II. R=E(above=4)below=k=1 f(wk)deltaxk, where subdivisions are at {0, 2, 3, 4, 6} and midpoints are used.

III.R=E(above=6)below=k=1 f(wk)deltaxk , where 6 equal subdivisions and right-hand endpoints are used.

I only
II only
III only
I, II, and III

6. Here is a graph of the function:

http://imagizer.imageshack.us/v2/800x600q90/826/zebj.jpg

Estimate the total area under this curve on the interval [0, 12] with a Riemann sum using 36 equal subdivisions and circumscribed rectangles. Hint: use symmetry to make this problem easier.

57.340
14.439
49.914
28.044

2. ## Re: 6 problems! Any help appreciated :)

Let's start with #1. You divide [0,1] into 10 equal sub-intervals [0,0.1],[0.1,0.2], ... ,[0.9,1]. Then you estimate the area under the curve from 0 to 0.1 (for example) by a rectangle with width 0.1 and height f(0.05) - this is the midpoint rule (0.05 is the midpoint of [0,0.1]). So you add up the areas from all 10 sub-intervals and that's the estimate of your integral.

If you show me your work, I could check it for you.

Numbers 2, 3, and 6 are similar, just with different methods of finding the height of the rectangles.

- Hollywood