Question

double integral problems help!

Answer 1

\[\int\limits \int\limits xy ^{2} / (x ^{2} + 1)\]

Answer 2

x from 0 to 1, y from -3 to 3

Answer 3

a quick calculation shows that the solution is 9

Answer 4

This is multivariate double\[\int\limits_{0}^{1}\int\limits_{-3}^{3}[ xy ^{2}/(x ^{2} +1)] dy dx\] integral integrated as follows:
first integrate the inner integral first ( i.e integral with respect to y).
In this case x will be considered as a constant.

\[\int\limits_{0}^{1}[ xy^{3}/3(x ^{2} +1)] dx\] Over the limits -3 to 3

Answer 5

= \[\int\limits_{0}^{1} 2*27*x/(3(x ^{2} + 1)) dx\]

Answer 6

Now complete the final integral with respect to x

we know that \[\int\limits_{?}^{?} x/(x ^{2} +1)dx = \ln (x ^{2} + 1)\]
If this is not clear we can prove that

Answer 7

so the final integrated values will be 2*27* (l n (x^2 +1)/3

Answer 8

substitue the limits of x between 0 and 1 as given

Answer 9

i.e. 54/3 (( ln(1 +1) - lg(0+1))

= 18((ln(2) + ln(1))
= 18 * 1 + 0
=18

Answer 10

Need to verify the arithmetic but that is how it is done

Answer 11

steps:

1. Integrate the inner integral with respect to y
And assumin that x is a constant.
2. Apply the y limits.
Now the integral should have only x as a variable.
3. Integrate the function from step 2 above with respect to x
4. Apply the x limits

Answer 12

slight correction

\[\int\limits_{?}^{?} x /(x ^{2} +1)dx = (1/2) * \ln (x ^{2} + 1)\]

which implies that the final answer should be divided by 2 ( i.e. 18/2)

Hencence the result is 9.


The final answer is 9