Evaluate the integral , where
, by considering D as a region of Type I and then as a region of Type II.
First, describe the region (D): [ D = {(x, y) : x \ge 0, y \ge 0, x + y \le 4}. ] This is a right triangle in the first quadrant with vertices at ((0,0)), ((4,0)), and ((0,4)). Type I (x-bounds constant, y-bounds variable): For Type I, (x) is the independent variable: (0 \le x \le 4), and for each (x), (y) ranges from (0) to (4 - x). [ \iint_D e^{2x + 3y} , dy , dx ________ _______ ____ _____ ________.
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Evaluate the integral , where
, by considering D as a region of Type I and then as a region of Type II.
If f is a function from set X to a set Y then is it possible ,
? where
and
In R we have a built-in data set "trees". A screenshot of the last four rows together with the R code to obtain it is given as follows. To get more detail about this data set you can run ?trees command on R console.
```
> tail(trees, 4)
Girth Height Volume
28 17.9 80 58.3
29 18.0 80 51.5
30 18.0 80 51.0
31 20.6 87 77.0
```
Note that all the three variables of this data set are numeric. So, assuming each row of this data set is a point in 3-dimension. Find the distances between the points corresponding to the and the
rows using the Manhattan and Chebyshev distance formula.
Find the equation of a line passing through points A(0, 1, 1) and B(1, 1, 0). Also, find the coordinates of a point on this line which is at a distance of 10 units from point A opposite to the side of point B.
(2 + 3 + 5)
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If be a function defined by
,
. Show that f is Riemann integrable using both definitions. Also, verify that the results of both definition match.
Evaluate the integral using beta and gamma functions.