(a) Let f x , y = x − 2 y , and as shown in Exercise 17, let the rectangle R = 0 , 2 × 0 , 2 be subdivided into 16 subrectangles. Take x k * , y k * to be the center of the k th rectangle, and approximate the double integral of f over R by the resulting Riemann sum. (b) Compare the result in part (a) to the exact value of the integral.
(a) Let f x , y = x − 2 y , and as shown in Exercise 17, let the rectangle R = 0 , 2 × 0 , 2 be subdivided into 16 subrectangles. Take x k * , y k * to be the center of the k th rectangle, and approximate the double integral of f over R by the resulting Riemann sum. (b) Compare the result in part (a) to the exact value of the integral.
(a) Let
f
x
,
y
=
x
−
2
y
,
and as shown in Exercise 17, let the rectangle
R
=
0
,
2
×
0
,
2
be subdivided into 16 subrectangles. Take
x
k
*
,
y
k
*
to be the center of the
k
th
rectangle, and approximate the double integral of
f
over
R
by the resulting Riemann sum.
(b) Compare the result in part (a) to the exact value of the integral.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
F) Evaluate the Gaussian integral
-10²
dr. (Ans: √/10)
f(x, y) = x²ex² and let R be the triangle bounded by the lines x = 5, x = y/2, and y = x
in the xy-plane.
R
(a) Express f f dA as a double integral in two different ways by filling in the values for the integrals
below. (For one of these it will be necessary to write the double integral as a sum of two integrals, as
indicated; for the other, it can be written as a single integral.)
fRf dA = få fd f(x,y) d y
where a = 0
d =
2x
And f f dA = √₂ fª f(x, y) d
R
where a =
, m =
and q =
, b =
, b =
d x
5
d
n =
+ fm f f(x,y) d
(b) Evaluate one of your integrals to find the value of ff dA.
√ Rf dA =
C = X
"
C =
, p =
d
and
d =
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